Rumus MA (2): \[ Y_t = e_t - \theta_1 e_{t-1} - \theta_2 e_{t-2} \] dengan \(\theta_1\) = 0.4 dan \(\theta_2\) = 0.6.
## [1] 0.131540851 -0.897909868 1.071705175 0.422106613 0.690736960
## [6] 1.501222304 2.311474041 -0.766756080 -0.375272150 -2.198788160
## [11] -0.770197848 -1.235168383 -1.884954880 -1.148267311 -0.165788735
## [16] -0.543028834 -0.698251506 -0.595470096 -1.361763778 -1.223980621
## [21] -0.872896857 0.169333233 -0.629372146 0.504724047 -1.080874546
## [26] 0.741720747 -0.434947966 -0.736240440 -0.898001000 -2.583414475
## [31] -1.946601852 0.405487512 -0.001458042 1.425295218 -0.992023400
## [36] 0.135076591 -0.756077274 -0.286595860 -0.890964944 -1.451956836
## [41] -1.119449938 -0.332941093 -0.149323883 0.900367374 -0.471636090
## [46] 0.291421594 -0.027813907 -0.360449808 0.298325813 1.103341959
## [51] 2.564457298 0.752839686 -0.349149879 -1.291486161 -0.809197685
## [56] -0.392505751 0.872787277 1.853672380 1.074961292 2.524086251
## [61] 2.143785091 2.242967913 2.051877771 0.367790380 0.884919174
## [66] -1.987698959 0.081619557 -1.399009877 -0.602541199 -0.573444638
## [71] 0.085263990 -0.098161217 -0.549525949 0.557429691 -0.441211933
## [76] -1.884837560 -1.408256548 -1.643425317 -0.199802210 2.034818368
## [81] 2.074401582 0.006957924 1.779372840 -0.841885624 1.362601273
## [86] -1.450510469 -0.907705729 -2.638564323 -0.122474746 -3.201485783
## [91] -1.104120758 -1.141750432 -1.392526292 -0.166637697 -1.946759635
## [96] -1.013813490 -1.976774025 -1.486695025 0.224089477 -1.855512321
## [101] -0.561068578 -0.999440381 -0.221545191 0.486811559 -0.036862815
## [106] 1.757290723 1.480190543 2.969062938 -0.326259776 -0.799197607
## [111] -0.808179007 -0.385273038 -0.348275098 -0.392513797 -0.063553296
## [116] 0.852222858 -0.496682531 -0.866590058 -0.905186937 -1.352029783
## [121] -0.713449120 0.331004139 -0.486937546 1.312935319 -1.237812436
## [126] 0.933383571 2.552545424 0.803360374 1.440802780 -0.446742946
## [131] 0.659236685 1.082173831 0.265199909 0.367789172 0.709282753
## [136] 1.385972415 0.802520637 -0.344503674 0.107120036 0.003476325
## [141] 1.392748099 0.779390228 -0.399078690 -1.094922224 -0.829390991
## [146] -1.864746160 -0.580276778 -0.478476828 0.356114144 1.850982956
## [151] -0.466901838 -1.164093416 0.137231410 1.168021742 3.261566866
## [156] 0.214870138 0.032633853 -1.860788790 -1.271352557 -0.825600567
## [161] 0.743515553 0.534708245 -0.005196627 0.202911024 -0.438505521
## [166] 1.111069126 -1.286172722 0.872794010 1.071890787 2.335583939
## [171] 0.940715791 -0.794767895 -1.429381201 0.145260459 -1.255209507
## [176] 0.938142085 -1.336776697 -0.115418604 -0.337151803 -0.660551833
## [181] 0.846697090 -0.042584924 1.938158720 2.904113441 0.118820814
## [186] 2.150886949 0.276600733 0.942818659 -0.463805594 1.786619578
## [191] 0.670057145 3.504060746 0.297509661 1.676409815 1.982706497
## [196] 1.634164958 1.189825747 -0.670367646 -0.652518236 -1.151377243
## [201] 0.148291269 -0.752651306 0.638619521 0.556678749 0.820734496
## [206] 0.804256928 0.552395492 1.165532369 -0.022253703 0.679273029
## [211] -0.342307569 -0.872145828 -1.478253847 -0.233706635 1.081041466
## [216] -0.788098167 0.819653163 -0.539672665 1.593565360 2.408443779
## [221] 0.221234597 2.239556708 1.617172807 1.181696461 0.784379850
## [226] -2.054508130 -0.569877189 -3.169308305 -0.651185696 -1.438204904
## [231] 0.072286160 -0.052327767 -0.096400102 -1.637654918 -2.208935588
## [236] -1.415831277 -0.444384665 0.850952959 1.501825451 1.038778453
## [241] -0.821503684 -0.975325820 -1.675504805 -1.246010938 1.860748958
## [246] -0.091680625 3.197791619 0.151798456 1.097527485 1.816289747
## [251] 0.956300101 0.770455741 0.294948362 -0.276326225 0.547203914
## [256] 0.016512894 1.018900124 1.300736390 2.087161732 -1.080885316
## [261] -1.642577893 -1.887157635 0.150056991 0.349669900 -0.797549589
## [266] -0.864764450 -1.211035093 -0.208354961 0.275473685 -1.066739767
## [271] -1.396369675 -1.512475371 0.406910154 0.214162508 0.575648631
## [276] -0.786718563 -0.674399863 -0.432780947 0.084417100 -0.397545750
## [281] -1.729390071 -1.769996549 -3.323988744 -2.402821052 -1.998021144
## [286] -0.912482741 -0.241228226 -0.802654706 -2.645049342 -1.464916821
## [291] 1.459673093 -0.056749942 0.358007119 1.119655842 -0.754682598
## [296] 1.260802819 -0.395196982 -0.532986680 -1.859035477 -3.405715799
## Time Series:
## Start = 1
## End = 300
## Frequency = 1
## [1] 1.071705175 0.422106613 0.690736960 1.501222304 2.311474041
## [6] -0.766756080 -0.375272150 -2.198788160 -0.770197848 -1.235168383
## [11] -1.884954880 -1.148267311 -0.165788735 -0.543028834 -0.698251506
## [16] -0.595470096 -1.361763778 -1.223980621 -0.872896857 0.169333233
## [21] -0.629372146 0.504724047 -1.080874546 0.741720747 -0.434947966
## [26] -0.736240440 -0.898001000 -2.583414475 -1.946601852 0.405487512
## [31] -0.001458042 1.425295218 -0.992023400 0.135076591 -0.756077274
## [36] -0.286595860 -0.890964944 -1.451956836 -1.119449938 -0.332941093
## [41] -0.149323883 0.900367374 -0.471636090 0.291421594 -0.027813907
## [46] -0.360449808 0.298325813 1.103341959 2.564457298 0.752839686
## [51] -0.349149879 -1.291486161 -0.809197685 -0.392505751 0.872787277
## [56] 1.853672380 1.074961292 2.524086251 2.143785091 2.242967913
## [61] 2.051877771 0.367790380 0.884919174 -1.987698959 0.081619557
## [66] -1.399009877 -0.602541199 -0.573444638 0.085263990 -0.098161217
## [71] -0.549525949 0.557429691 -0.441211933 -1.884837560 -1.408256548
## [76] -1.643425317 -0.199802210 2.034818368 2.074401582 0.006957924
## [81] 1.779372840 -0.841885624 1.362601273 -1.450510469 -0.907705729
## [86] -2.638564323 -0.122474746 -3.201485783 -1.104120758 -1.141750432
## [91] -1.392526292 -0.166637697 -1.946759635 -1.013813490 -1.976774025
## [96] -1.486695025 0.224089477 -1.855512321 -0.561068578 -0.999440381
## [101] -0.221545191 0.486811559 -0.036862815 1.757290723 1.480190543
## [106] 2.969062938 -0.326259776 -0.799197607 -0.808179007 -0.385273038
## [111] -0.348275098 -0.392513797 -0.063553296 0.852222858 -0.496682531
## [116] -0.866590058 -0.905186937 -1.352029783 -0.713449120 0.331004139
## [121] -0.486937546 1.312935319 -1.237812436 0.933383571 2.552545424
## [126] 0.803360374 1.440802780 -0.446742946 0.659236685 1.082173831
## [131] 0.265199909 0.367789172 0.709282753 1.385972415 0.802520637
## [136] -0.344503674 0.107120036 0.003476325 1.392748099 0.779390228
## [141] -0.399078690 -1.094922224 -0.829390991 -1.864746160 -0.580276778
## [146] -0.478476828 0.356114144 1.850982956 -0.466901838 -1.164093416
## [151] 0.137231410 1.168021742 3.261566866 0.214870138 0.032633853
## [156] -1.860788790 -1.271352557 -0.825600567 0.743515553 0.534708245
## [161] -0.005196627 0.202911024 -0.438505521 1.111069126 -1.286172722
## [166] 0.872794010 1.071890787 2.335583939 0.940715791 -0.794767895
## [171] -1.429381201 0.145260459 -1.255209507 0.938142085 -1.336776697
## [176] -0.115418604 -0.337151803 -0.660551833 0.846697090 -0.042584924
## [181] 1.938158720 2.904113441 0.118820814 2.150886949 0.276600733
## [186] 0.942818659 -0.463805594 1.786619578 0.670057145 3.504060746
## [191] 0.297509661 1.676409815 1.982706497 1.634164958 1.189825747
## [196] -0.670367646 -0.652518236 -1.151377243 0.148291269 -0.752651306
## [201] 0.638619521 0.556678749 0.820734496 0.804256928 0.552395492
## [206] 1.165532369 -0.022253703 0.679273029 -0.342307569 -0.872145828
## [211] -1.478253847 -0.233706635 1.081041466 -0.788098167 0.819653163
## [216] -0.539672665 1.593565360 2.408443779 0.221234597 2.239556708
## [221] 1.617172807 1.181696461 0.784379850 -2.054508130 -0.569877189
## [226] -3.169308305 -0.651185696 -1.438204904 0.072286160 -0.052327767
## [231] -0.096400102 -1.637654918 -2.208935588 -1.415831277 -0.444384665
## [236] 0.850952959 1.501825451 1.038778453 -0.821503684 -0.975325820
## [241] -1.675504805 -1.246010938 1.860748958 -0.091680625 3.197791619
## [246] 0.151798456 1.097527485 1.816289747 0.956300101 0.770455741
## [251] 0.294948362 -0.276326225 0.547203914 0.016512894 1.018900124
## [256] 1.300736390 2.087161732 -1.080885316 -1.642577893 -1.887157635
## [261] 0.150056991 0.349669900 -0.797549589 -0.864764450 -1.211035093
## [266] -0.208354961 0.275473685 -1.066739767 -1.396369675 -1.512475371
## [271] 0.406910154 0.214162508 0.575648631 -0.786718563 -0.674399863
## [276] -0.432780947 0.084417100 -0.397545750 -1.729390071 -1.769996549
## [281] -3.323988744 -2.402821052 -1.998021144 -0.912482741 -0.241228226
## [286] -0.802654706 -2.645049342 -1.464916821 1.459673093 -0.056749942
## [291] 0.358007119 1.119655842 -0.754682598 1.260802819 -0.395196982
## [296] -0.532986680 -1.859035477 -3.405715799 -2.114086037 -0.915226670
Berdasarkan plot time series, terlihat bahwa data MA(2) yang dibangkitkan stasioner dalam rataan.
Berdasarkan plot ACF tersebut, terlihat bahwa plot ACF cuts off di lag pertama, tidak sejalan dengan teori yang ada.
Berdasarkan plot PACF tersebut, terlihat bahwa plot PACF cenderung tails off dan membentuk gelombang sinus.
## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x o o o o o o o o o o o o
## 1 x x x o o o o o o o o o o o
## 2 x x o x o o o o o o o o o o
## 3 x x o x o o o o o o o o o o
## 4 x x x x o o o o o o o o o o
## 5 x x x x o o o o o o o o o o
## 6 x x x x o o o o o o o o o o
## 7 x x x o o o o o o o o o o o
Berdasarkan pola segitiga nol pada plot EACF, terlihat bahwa segitiga nol pertama berada pada ordo AR(0) dan ordo MA(2).
## [1] 0.422106613 0.690736960 1.501222304 2.311474041 -0.766756080
## [6] -0.375272150 -2.198788160 -0.770197848 -1.235168383 -1.884954880
## [11] -1.148267311 -0.165788735 -0.543028834 -0.698251506 -0.595470096
## [16] -1.361763778 -1.223980621 -0.872896857 0.169333233 -0.629372146
## [21] 0.504724047 -1.080874546 0.741720747 -0.434947966 -0.736240440
## [26] -0.898001000 -2.583414475 -1.946601852 0.405487512 -0.001458042
## [31] 1.425295218 -0.992023400 0.135076591 -0.756077274 -0.286595860
## [36] -0.890964944 -1.451956836 -1.119449938 -0.332941093 -0.149323883
## [41] 0.900367374 -0.471636090 0.291421594 -0.027813907 -0.360449808
## [46] 0.298325813 1.103341959 2.564457298 0.752839686 -0.349149879
## [51] -1.291486161 -0.809197685 -0.392505751 0.872787277 1.853672380
## [56] 1.074961292 2.524086251 2.143785091 2.242967913 2.051877771
## [61] 0.367790380 0.884919174 -1.987698959 0.081619557 -1.399009877
## [66] -0.602541199 -0.573444638 0.085263990 -0.098161217 -0.549525949
## [71] 0.557429691 -0.441211933 -1.884837560 -1.408256548 -1.643425317
## [76] -0.199802210 2.034818368 2.074401582 0.006957924 1.779372840
## [81] -0.841885624 1.362601273 -1.450510469 -0.907705729 -2.638564323
## [86] -0.122474746 -3.201485783 -1.104120758 -1.141750432 -1.392526292
## [91] -0.166637697 -1.946759635 -1.013813490 -1.976774025 -1.486695025
## [96] 0.224089477 -1.855512321 -0.561068578 -0.999440381 -0.221545191
## [101] 0.486811559 -0.036862815 1.757290723 1.480190543 2.969062938
## [106] -0.326259776 -0.799197607 -0.808179007 -0.385273038 -0.348275098
## [111] -0.392513797 -0.063553296 0.852222858 -0.496682531 -0.866590058
## [116] -0.905186937 -1.352029783 -0.713449120 0.331004139 -0.486937546
## [121] 1.312935319 -1.237812436 0.933383571 2.552545424 0.803360374
## [126] 1.440802780 -0.446742946 0.659236685 1.082173831 0.265199909
## [131] 0.367789172 0.709282753 1.385972415 0.802520637 -0.344503674
## [136] 0.107120036 0.003476325 1.392748099 0.779390228 -0.399078690
## [141] -1.094922224 -0.829390991 -1.864746160 -0.580276778 -0.478476828
## [146] 0.356114144 1.850982956 -0.466901838 -1.164093416 0.137231410
## [151] 1.168021742 3.261566866 0.214870138 0.032633853 -1.860788790
## [156] -1.271352557 -0.825600567 0.743515553 0.534708245 -0.005196627
## [161] 0.202911024 -0.438505521 1.111069126 -1.286172722 0.872794010
## [166] 1.071890787 2.335583939 0.940715791 -0.794767895 -1.429381201
## [171] 0.145260459 -1.255209507 0.938142085 -1.336776697 -0.115418604
## [176] -0.337151803 -0.660551833 0.846697090 -0.042584924 1.938158720
## [181] 2.904113441 0.118820814 2.150886949 0.276600733 0.942818659
## [186] -0.463805594 1.786619578 0.670057145 3.504060746 0.297509661
## [191] 1.676409815 1.982706497 1.634164958 1.189825747 -0.670367646
## [196] -0.652518236 -1.151377243 0.148291269 -0.752651306 0.638619521
## [201] 0.556678749 0.820734496 0.804256928 0.552395492 1.165532369
## [206] -0.022253703 0.679273029 -0.342307569 -0.872145828 -1.478253847
## [211] -0.233706635 1.081041466 -0.788098167 0.819653163 -0.539672665
## [216] 1.593565360 2.408443779 0.221234597 2.239556708 1.617172807
## [221] 1.181696461 0.784379850 -2.054508130 -0.569877189 -3.169308305
## [226] -0.651185696 -1.438204904 0.072286160 -0.052327767 -0.096400102
## [231] -1.637654918 -2.208935588 -1.415831277 -0.444384665 0.850952959
## [236] 1.501825451 1.038778453 -0.821503684 -0.975325820 -1.675504805
## [241] -1.246010938 1.860748958 -0.091680625 3.197791619 0.151798456
## [246] 1.097527485 1.816289747 0.956300101 0.770455741 0.294948362
## [251] -0.276326225 0.547203914 0.016512894 1.018900124 1.300736390
## [256] 2.087161732 -1.080885316 -1.642577893 -1.887157635 0.150056991
## [261] 0.349669900 -0.797549589 -0.864764450 -1.211035093 -0.208354961
## [266] 0.275473685 -1.066739767 -1.396369675 -1.512475371 0.406910154
## [271] 0.214162508 0.575648631 -0.786718563 -0.674399863 -0.432780947
## [276] 0.084417100 -0.397545750 -1.729390071 -1.769996549 -3.323988744
## [281] -2.402821052 -1.998021144 -0.912482741 -0.241228226 -0.802654706
## [286] -2.645049342 -1.464916821 1.459673093 -0.056749942 0.358007119
## [291] 1.119655842 -0.754682598 1.260802819 -0.395196982 -0.532986680
## [296] -1.859035477 -3.405715799
## [1] 1.071705175 0.422106613 0.690736960 1.501222304 2.311474041
## [6] -0.766756080 -0.375272150 -2.198788160 -0.770197848 -1.235168383
## [11] -1.884954880 -1.148267311 -0.165788735 -0.543028834 -0.698251506
## [16] -0.595470096 -1.361763778 -1.223980621 -0.872896857 0.169333233
## [21] -0.629372146 0.504724047 -1.080874546 0.741720747 -0.434947966
## [26] -0.736240440 -0.898001000 -2.583414475 -1.946601852 0.405487512
## [31] -0.001458042 1.425295218 -0.992023400 0.135076591 -0.756077274
## [36] -0.286595860 -0.890964944 -1.451956836 -1.119449938 -0.332941093
## [41] -0.149323883 0.900367374 -0.471636090 0.291421594 -0.027813907
## [46] -0.360449808 0.298325813 1.103341959 2.564457298 0.752839686
## [51] -0.349149879 -1.291486161 -0.809197685 -0.392505751 0.872787277
## [56] 1.853672380 1.074961292 2.524086251 2.143785091 2.242967913
## [61] 2.051877771 0.367790380 0.884919174 -1.987698959 0.081619557
## [66] -1.399009877 -0.602541199 -0.573444638 0.085263990 -0.098161217
## [71] -0.549525949 0.557429691 -0.441211933 -1.884837560 -1.408256548
## [76] -1.643425317 -0.199802210 2.034818368 2.074401582 0.006957924
## [81] 1.779372840 -0.841885624 1.362601273 -1.450510469 -0.907705729
## [86] -2.638564323 -0.122474746 -3.201485783 -1.104120758 -1.141750432
## [91] -1.392526292 -0.166637697 -1.946759635 -1.013813490 -1.976774025
## [96] -1.486695025 0.224089477 -1.855512321 -0.561068578 -0.999440381
## [101] -0.221545191 0.486811559 -0.036862815 1.757290723 1.480190543
## [106] 2.969062938 -0.326259776 -0.799197607 -0.808179007 -0.385273038
## [111] -0.348275098 -0.392513797 -0.063553296 0.852222858 -0.496682531
## [116] -0.866590058 -0.905186937 -1.352029783 -0.713449120 0.331004139
## [121] -0.486937546 1.312935319 -1.237812436 0.933383571 2.552545424
## [126] 0.803360374 1.440802780 -0.446742946 0.659236685 1.082173831
## [131] 0.265199909 0.367789172 0.709282753 1.385972415 0.802520637
## [136] -0.344503674 0.107120036 0.003476325 1.392748099 0.779390228
## [141] -0.399078690 -1.094922224 -0.829390991 -1.864746160 -0.580276778
## [146] -0.478476828 0.356114144 1.850982956 -0.466901838 -1.164093416
## [151] 0.137231410 1.168021742 3.261566866 0.214870138 0.032633853
## [156] -1.860788790 -1.271352557 -0.825600567 0.743515553 0.534708245
## [161] -0.005196627 0.202911024 -0.438505521 1.111069126 -1.286172722
## [166] 0.872794010 1.071890787 2.335583939 0.940715791 -0.794767895
## [171] -1.429381201 0.145260459 -1.255209507 0.938142085 -1.336776697
## [176] -0.115418604 -0.337151803 -0.660551833 0.846697090 -0.042584924
## [181] 1.938158720 2.904113441 0.118820814 2.150886949 0.276600733
## [186] 0.942818659 -0.463805594 1.786619578 0.670057145 3.504060746
## [191] 0.297509661 1.676409815 1.982706497 1.634164958 1.189825747
## [196] -0.670367646 -0.652518236 -1.151377243 0.148291269 -0.752651306
## [201] 0.638619521 0.556678749 0.820734496 0.804256928 0.552395492
## [206] 1.165532369 -0.022253703 0.679273029 -0.342307569 -0.872145828
## [211] -1.478253847 -0.233706635 1.081041466 -0.788098167 0.819653163
## [216] -0.539672665 1.593565360 2.408443779 0.221234597 2.239556708
## [221] 1.617172807 1.181696461 0.784379850 -2.054508130 -0.569877189
## [226] -3.169308305 -0.651185696 -1.438204904 0.072286160 -0.052327767
## [231] -0.096400102 -1.637654918 -2.208935588 -1.415831277 -0.444384665
## [236] 0.850952959 1.501825451 1.038778453 -0.821503684 -0.975325820
## [241] -1.675504805 -1.246010938 1.860748958 -0.091680625 3.197791619
## [246] 0.151798456 1.097527485 1.816289747 0.956300101 0.770455741
## [251] 0.294948362 -0.276326225 0.547203914 0.016512894 1.018900124
## [256] 1.300736390 2.087161732 -1.080885316 -1.642577893 -1.887157635
## [261] 0.150056991 0.349669900 -0.797549589 -0.864764450 -1.211035093
## [266] -0.208354961 0.275473685 -1.066739767 -1.396369675 -1.512475371
## [271] 0.406910154 0.214162508 0.575648631 -0.786718563 -0.674399863
## [276] -0.432780947 0.084417100 -0.397545750 -1.729390071 -1.769996549
## [281] -3.323988744 -2.402821052 -1.998021144 -0.912482741 -0.241228226
## [286] -0.802654706 -2.645049342 -1.464916821 1.459673093 -0.056749942
## [291] 0.358007119 1.119655842 -0.754682598 1.260802819 -0.395196982
## [296] -0.532986680 -1.859035477
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-1}\).
## [1] -0.897909868 1.071705175 0.422106613 0.690736960 1.501222304
## [6] 2.311474041 -0.766756080 -0.375272150 -2.198788160 -0.770197848
## [11] -1.235168383 -1.884954880 -1.148267311 -0.165788735 -0.543028834
## [16] -0.698251506 -0.595470096 -1.361763778 -1.223980621 -0.872896857
## [21] 0.169333233 -0.629372146 0.504724047 -1.080874546 0.741720747
## [26] -0.434947966 -0.736240440 -0.898001000 -2.583414475 -1.946601852
## [31] 0.405487512 -0.001458042 1.425295218 -0.992023400 0.135076591
## [36] -0.756077274 -0.286595860 -0.890964944 -1.451956836 -1.119449938
## [41] -0.332941093 -0.149323883 0.900367374 -0.471636090 0.291421594
## [46] -0.027813907 -0.360449808 0.298325813 1.103341959 2.564457298
## [51] 0.752839686 -0.349149879 -1.291486161 -0.809197685 -0.392505751
## [56] 0.872787277 1.853672380 1.074961292 2.524086251 2.143785091
## [61] 2.242967913 2.051877771 0.367790380 0.884919174 -1.987698959
## [66] 0.081619557 -1.399009877 -0.602541199 -0.573444638 0.085263990
## [71] -0.098161217 -0.549525949 0.557429691 -0.441211933 -1.884837560
## [76] -1.408256548 -1.643425317 -0.199802210 2.034818368 2.074401582
## [81] 0.006957924 1.779372840 -0.841885624 1.362601273 -1.450510469
## [86] -0.907705729 -2.638564323 -0.122474746 -3.201485783 -1.104120758
## [91] -1.141750432 -1.392526292 -0.166637697 -1.946759635 -1.013813490
## [96] -1.976774025 -1.486695025 0.224089477 -1.855512321 -0.561068578
## [101] -0.999440381 -0.221545191 0.486811559 -0.036862815 1.757290723
## [106] 1.480190543 2.969062938 -0.326259776 -0.799197607 -0.808179007
## [111] -0.385273038 -0.348275098 -0.392513797 -0.063553296 0.852222858
## [116] -0.496682531 -0.866590058 -0.905186937 -1.352029783 -0.713449120
## [121] 0.331004139 -0.486937546 1.312935319 -1.237812436 0.933383571
## [126] 2.552545424 0.803360374 1.440802780 -0.446742946 0.659236685
## [131] 1.082173831 0.265199909 0.367789172 0.709282753 1.385972415
## [136] 0.802520637 -0.344503674 0.107120036 0.003476325 1.392748099
## [141] 0.779390228 -0.399078690 -1.094922224 -0.829390991 -1.864746160
## [146] -0.580276778 -0.478476828 0.356114144 1.850982956 -0.466901838
## [151] -1.164093416 0.137231410 1.168021742 3.261566866 0.214870138
## [156] 0.032633853 -1.860788790 -1.271352557 -0.825600567 0.743515553
## [161] 0.534708245 -0.005196627 0.202911024 -0.438505521 1.111069126
## [166] -1.286172722 0.872794010 1.071890787 2.335583939 0.940715791
## [171] -0.794767895 -1.429381201 0.145260459 -1.255209507 0.938142085
## [176] -1.336776697 -0.115418604 -0.337151803 -0.660551833 0.846697090
## [181] -0.042584924 1.938158720 2.904113441 0.118820814 2.150886949
## [186] 0.276600733 0.942818659 -0.463805594 1.786619578 0.670057145
## [191] 3.504060746 0.297509661 1.676409815 1.982706497 1.634164958
## [196] 1.189825747 -0.670367646 -0.652518236 -1.151377243 0.148291269
## [201] -0.752651306 0.638619521 0.556678749 0.820734496 0.804256928
## [206] 0.552395492 1.165532369 -0.022253703 0.679273029 -0.342307569
## [211] -0.872145828 -1.478253847 -0.233706635 1.081041466 -0.788098167
## [216] 0.819653163 -0.539672665 1.593565360 2.408443779 0.221234597
## [221] 2.239556708 1.617172807 1.181696461 0.784379850 -2.054508130
## [226] -0.569877189 -3.169308305 -0.651185696 -1.438204904 0.072286160
## [231] -0.052327767 -0.096400102 -1.637654918 -2.208935588 -1.415831277
## [236] -0.444384665 0.850952959 1.501825451 1.038778453 -0.821503684
## [241] -0.975325820 -1.675504805 -1.246010938 1.860748958 -0.091680625
## [246] 3.197791619 0.151798456 1.097527485 1.816289747 0.956300101
## [251] 0.770455741 0.294948362 -0.276326225 0.547203914 0.016512894
## [256] 1.018900124 1.300736390 2.087161732 -1.080885316 -1.642577893
## [261] -1.887157635 0.150056991 0.349669900 -0.797549589 -0.864764450
## [266] -1.211035093 -0.208354961 0.275473685 -1.066739767 -1.396369675
## [271] -1.512475371 0.406910154 0.214162508 0.575648631 -0.786718563
## [276] -0.674399863 -0.432780947 0.084417100 -0.397545750 -1.729390071
## [281] -1.769996549 -3.323988744 -2.402821052 -1.998021144 -0.912482741
## [286] -0.241228226 -0.802654706 -2.645049342 -1.464916821 1.459673093
## [291] -0.056749942 0.358007119 1.119655842 -0.754682598 1.260802819
## [296] -0.395196982 -0.532986680
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-2}\).
## [1] 0.131540851 -0.897909868 1.071705175 0.422106613 0.690736960
## [6] 1.501222304 2.311474041 -0.766756080 -0.375272150 -2.198788160
## [11] -0.770197848 -1.235168383 -1.884954880 -1.148267311 -0.165788735
## [16] -0.543028834 -0.698251506 -0.595470096 -1.361763778 -1.223980621
## [21] -0.872896857 0.169333233 -0.629372146 0.504724047 -1.080874546
## [26] 0.741720747 -0.434947966 -0.736240440 -0.898001000 -2.583414475
## [31] -1.946601852 0.405487512 -0.001458042 1.425295218 -0.992023400
## [36] 0.135076591 -0.756077274 -0.286595860 -0.890964944 -1.451956836
## [41] -1.119449938 -0.332941093 -0.149323883 0.900367374 -0.471636090
## [46] 0.291421594 -0.027813907 -0.360449808 0.298325813 1.103341959
## [51] 2.564457298 0.752839686 -0.349149879 -1.291486161 -0.809197685
## [56] -0.392505751 0.872787277 1.853672380 1.074961292 2.524086251
## [61] 2.143785091 2.242967913 2.051877771 0.367790380 0.884919174
## [66] -1.987698959 0.081619557 -1.399009877 -0.602541199 -0.573444638
## [71] 0.085263990 -0.098161217 -0.549525949 0.557429691 -0.441211933
## [76] -1.884837560 -1.408256548 -1.643425317 -0.199802210 2.034818368
## [81] 2.074401582 0.006957924 1.779372840 -0.841885624 1.362601273
## [86] -1.450510469 -0.907705729 -2.638564323 -0.122474746 -3.201485783
## [91] -1.104120758 -1.141750432 -1.392526292 -0.166637697 -1.946759635
## [96] -1.013813490 -1.976774025 -1.486695025 0.224089477 -1.855512321
## [101] -0.561068578 -0.999440381 -0.221545191 0.486811559 -0.036862815
## [106] 1.757290723 1.480190543 2.969062938 -0.326259776 -0.799197607
## [111] -0.808179007 -0.385273038 -0.348275098 -0.392513797 -0.063553296
## [116] 0.852222858 -0.496682531 -0.866590058 -0.905186937 -1.352029783
## [121] -0.713449120 0.331004139 -0.486937546 1.312935319 -1.237812436
## [126] 0.933383571 2.552545424 0.803360374 1.440802780 -0.446742946
## [131] 0.659236685 1.082173831 0.265199909 0.367789172 0.709282753
## [136] 1.385972415 0.802520637 -0.344503674 0.107120036 0.003476325
## [141] 1.392748099 0.779390228 -0.399078690 -1.094922224 -0.829390991
## [146] -1.864746160 -0.580276778 -0.478476828 0.356114144 1.850982956
## [151] -0.466901838 -1.164093416 0.137231410 1.168021742 3.261566866
## [156] 0.214870138 0.032633853 -1.860788790 -1.271352557 -0.825600567
## [161] 0.743515553 0.534708245 -0.005196627 0.202911024 -0.438505521
## [166] 1.111069126 -1.286172722 0.872794010 1.071890787 2.335583939
## [171] 0.940715791 -0.794767895 -1.429381201 0.145260459 -1.255209507
## [176] 0.938142085 -1.336776697 -0.115418604 -0.337151803 -0.660551833
## [181] 0.846697090 -0.042584924 1.938158720 2.904113441 0.118820814
## [186] 2.150886949 0.276600733 0.942818659 -0.463805594 1.786619578
## [191] 0.670057145 3.504060746 0.297509661 1.676409815 1.982706497
## [196] 1.634164958 1.189825747 -0.670367646 -0.652518236 -1.151377243
## [201] 0.148291269 -0.752651306 0.638619521 0.556678749 0.820734496
## [206] 0.804256928 0.552395492 1.165532369 -0.022253703 0.679273029
## [211] -0.342307569 -0.872145828 -1.478253847 -0.233706635 1.081041466
## [216] -0.788098167 0.819653163 -0.539672665 1.593565360 2.408443779
## [221] 0.221234597 2.239556708 1.617172807 1.181696461 0.784379850
## [226] -2.054508130 -0.569877189 -3.169308305 -0.651185696 -1.438204904
## [231] 0.072286160 -0.052327767 -0.096400102 -1.637654918 -2.208935588
## [236] -1.415831277 -0.444384665 0.850952959 1.501825451 1.038778453
## [241] -0.821503684 -0.975325820 -1.675504805 -1.246010938 1.860748958
## [246] -0.091680625 3.197791619 0.151798456 1.097527485 1.816289747
## [251] 0.956300101 0.770455741 0.294948362 -0.276326225 0.547203914
## [256] 0.016512894 1.018900124 1.300736390 2.087161732 -1.080885316
## [261] -1.642577893 -1.887157635 0.150056991 0.349669900 -0.797549589
## [266] -0.864764450 -1.211035093 -0.208354961 0.275473685 -1.066739767
## [271] -1.396369675 -1.512475371 0.406910154 0.214162508 0.575648631
## [276] -0.786718563 -0.674399863 -0.432780947 0.084417100 -0.397545750
## [281] -1.729390071 -1.769996549 -3.323988744 -2.402821052 -1.998021144
## [286] -0.912482741 -0.241228226 -0.802654706 -2.645049342 -1.464916821
## [291] 1.459673093 -0.056749942 0.358007119 1.119655842 -0.754682598
## [296] 1.260802819 -0.395196982
Berdasarkan scatterplot tersebut, terlihat bahwa cenderung tidak terdapat hubungan antara \(Y_t\) dan \(Y_{t-3}\).
## [1] 0.4301445
Nalai korelasi antara \(Y_t\) dan \(Y_{t-1}\) dari hasil simulasi mendekati perhitungan teoritis yaitu:
\[ \rho_{1} = \frac{-\theta_1 + \theta_1\theta_2}{1 + (-\theta_1)^2 + (-\theta_2)^2} = \frac{-(-0.4) + (-0.4)(-0.6)}{1 + (-0.4)^2 + (-0.6)^2} \approx 0.421 \]
## [1] 0.3820866
Nilai korelasi antara \(Y_t\) dan \(Y_{t-2}\) dari hasil simulasi mendekati perhitungan teoritis yaitu:
\[ \rho_{2} = \frac{-\theta_2}{1 + (-\theta_1)^2 + (-\theta_2)^2} = \frac{-(-0.6)}{1 + (-0.4)^2 + (-0.6)^2} \approx 0.395 \]
## [1] 0.05579518
Nilai korelasi antara \(Y_t\) dan \(Y_{t-3}\) dari hasil simulasi kurang mendekati teori yang ada yaitu 0.
Kesimpulan: Meskipun nilai autokorelasi empiris cukup dekat dengan nilai teoritis, masih terdapat perbedaan pola, misalnya pada \(\rho_3\) yang secara teoritis seharusnya bernilai nol. Ketidaksesuaian ini umumnya disebabkan oleh keterbatasan sampel, fluktuasi estimasi ACF, atau parameter yang relatif kecil sehingga kontribusi lag tertentu tampak lemah. Dengan demikian, model MA(2) tetap konsisten dengan struktur teoritis, sedangkan perbedaan kecil yang muncul lebih disebabkan oleh faktor acak dalam proses estimasi, bukan karena kesalahan dalam pemilihan model.
Rumus AR (2): \[ Y_t = \phi_1 Y_{t-1} + \phi_2 Y_{t-2} + e_t \] dengan \(\phi_1\) = 0.5 dan \(\phi_2\) = 0.2.
## [1] 0.131540851 -0.897909868 0.929297848 0.705141640 0.249970706
## [6] 1.630575045 2.804006969 -0.632864527 -0.349744995 -0.845998100
## [11] -0.688856343 -1.343701238 -2.245002413 -1.467312487 -0.456788733
## [16] -1.309590275 -1.564831822 -0.839692440 -1.685225610 -1.976351619
## [21] -1.240349843 -0.300580252 -1.364600891 -0.280115127 -1.099028358
## [26] 0.133222967 -0.472015853 -1.261331210 -1.010989626 -2.595626463
## [31] -2.539914335 0.135085568 -0.587623429 0.062873138 -1.121668605
## [36] -0.196751533 -0.598041725 -0.725739020 -1.053296316 -1.663004703
## [41] -1.422636980 -0.629945133 -0.686128345 0.217580950 -0.722747914
## [46] -0.160691574 0.101011878 -0.566744798 0.073656410 1.243158084
## [51] 2.470800226 0.711268152 -0.290944268 -0.374751162 -0.221706023
## [56] -0.364932764 0.703316452 1.867772149 0.955796044 2.469579745
## [61] 2.993735248 2.635760644 2.769842614 1.551591476 1.846947073
## [66] -0.744491076 0.559766909 -0.306104106 -0.806377951 -0.469489115
## [71] 0.150443138 -0.332375194 -0.888023314 0.535932096 -0.318263458
## [76] -2.401184208 -1.487630879 -1.368579826 -0.989775631 1.356114931
## [81] 1.709392162 -0.633670165 1.770677990 0.274219021 1.000219974
## [86] -0.808515059 -0.871916643 -2.151063102 -0.350407053 -3.234663999
## [91] -2.279539723 -1.114053857 -2.319244900 -1.430136190 -2.322809507
## [96] -1.975066728 -2.531479937 -2.399113628 -0.538788396 -2.628538819
## [101] -1.931605552 -1.159526399 -1.014676079 -0.432185537 -0.549558850
## [106] 1.264051868 1.430515867 2.598609523 0.061890650 -0.617448193
## [111] 0.276831567 0.101251347 -0.620719712 -0.443733826 0.087912564
## [116] 0.726037551 -0.684766508 -1.100115503 -0.591801408 -1.364273590
## [121] -1.231732070 0.123787254 -0.817652540 0.574641597 -1.117586460
## [126] 0.410843111 2.937392529 0.659205141 0.941254795 0.681051840
## [131] 1.142075967 1.496952499 0.558230240 0.639936785 1.367521449
## [136] 1.786498994 1.017829854 -0.003569923 0.746169264 0.679962460
## [141] 1.432293504 1.069743017 -0.230637043 -0.706067505 -0.275492766
## [146] -1.710354850 -0.992195806 -0.425040791 -0.170944748 1.336619651
## [151] -0.579629920 -1.605344237 0.580025068 1.487228085 2.614698261
## [156] 0.206662565 0.165110137 -0.713557580 -0.983434777 -0.693691169
## [161] 0.619494728 0.276063826 -0.483104007 0.251552914 -0.137485807
## [166] 0.896558099 -1.131402124 0.558283920 1.678193384 2.069400658
## [171] 0.888396815 -0.415100789 -0.660815187 0.748117966 -1.099365355
## [176] 0.377660091 -0.874150612 -0.606350362 -0.211369725 -0.847276755
## [181] 0.468932148 0.020681103 1.499110867 3.126409313 0.195770249
## [186] 2.117325973 1.817108123 1.150626089 0.115911690 2.512665732
## [191] 1.553432269 3.338992399 1.289444828 2.029379968 3.382984512
## [196] 2.457255539 1.684852166 0.535726173 0.403799194 -0.283028070
## [201] 0.444986829 -0.433816175 0.447145774 0.823288188 0.702182433
## [206] 0.827674987 0.861233509 1.451744630 0.349452445 0.850459079
## [211] 0.327858831 -0.702450476 -1.248961917 0.008528714 1.104098600
## [216] -1.238293635 0.328369561 0.161403931 1.205972092 2.472945399
## [221] 0.328072560 2.255412751 2.861535473 1.438188479 1.252317156
## [226] -0.858881100 -0.016332851 -2.350702233 -1.059086108 -1.183245283
## [231] -0.729447937 -0.573160493 -0.584550104 -2.000657480 -2.597426055
## [236] -1.566350686 -0.911930377 -0.154070741 0.761900065 0.611286982
## [241] -1.080733526 -0.934623033 -1.229103342 -1.519320967 1.469792754
## [246] -0.220051190 2.156993581 0.787710783 0.837655525 2.535764561
## [251] 1.600514779 0.836188700 1.122573472 0.581180746 0.890620773
## [256] 0.516259386 1.247959408 1.730458037 2.313668714 -0.659434350
## [261] -1.363925887 -0.804814278 0.369415933 -0.049849610 -1.345918269
## [266] -0.945586014 -1.011022981 -0.637770952 -0.106931692 -1.447537800
## [271] -1.883412836 -1.528475651 0.067848989 -0.362758876 -0.281027342
## [276] -0.899903984 -0.837891287 -0.486910734 -0.172382511 -0.743789022
## [281] -2.054865504 -1.950659938 -3.411407367 -3.223933375 -2.625967645
## [286] -2.060706822 -1.556568946 -1.930909946 -3.625011258 -2.280016976
## [291] 1.036139594 -1.106346059 -1.261263207 1.334929741 -0.665011734
## [296] 0.315274152 0.125254497 -0.676028950 -1.911686502 -3.376249052
## Time Series:
## Start = 1
## End = 200
## Frequency = 1
## [1] 0.133295622 -0.471960465 -1.261288985 -1.010957436 -2.595601924
## [6] -2.539895627 0.135099830 -0.587612557 0.062881427 -1.121662286
## [11] -0.196746715 -0.598038053 -0.725736220 -1.053294182 -1.663003076
## [16] -1.422635739 -0.629944187 -0.686127624 0.217581499 -0.722747495
## [21] -0.160691255 0.101012122 -0.566744612 0.073656551 1.243158192
## [26] 2.470800309 0.711268215 -0.290944221 -0.374751126 -0.221705995
## [31] -0.364932743 0.703316468 1.867772162 0.955796053 2.469579753
## [36] 2.993735253 2.635760648 2.769842617 1.551591479 1.846947075
## [41] -0.744491075 0.559766910 -0.306104105 -0.806377951 -0.469489115
## [46] 0.150443138 -0.332375194 -0.888023314 0.535932097 -0.318263458
## [51] -2.401184208 -1.487630879 -1.368579826 -0.989775631 1.356114931
## [56] 1.709392163 -0.633670165 1.770677990 0.274219021 1.000219974
## [61] -0.808515059 -0.871916643 -2.151063102 -0.350407053 -3.234663999
## [66] -2.279539723 -1.114053857 -2.319244900 -1.430136190 -2.322809507
## [71] -1.975066728 -2.531479937 -2.399113628 -0.538788396 -2.628538819
## [76] -1.931605552 -1.159526399 -1.014676079 -0.432185537 -0.549558850
## [81] 1.264051868 1.430515867 2.598609523 0.061890650 -0.617448193
## [86] 0.276831567 0.101251347 -0.620719712 -0.443733826 0.087912564
## [91] 0.726037551 -0.684766508 -1.100115503 -0.591801408 -1.364273590
## [96] -1.231732070 0.123787254 -0.817652540 0.574641597 -1.117586460
## [101] 0.410843111 2.937392529 0.659205141 0.941254795 0.681051840
## [106] 1.142075967 1.496952499 0.558230240 0.639936785 1.367521449
## [111] 1.786498994 1.017829854 -0.003569923 0.746169264 0.679962460
## [116] 1.432293504 1.069743017 -0.230637043 -0.706067505 -0.275492766
## [121] -1.710354850 -0.992195806 -0.425040791 -0.170944748 1.336619651
## [126] -0.579629920 -1.605344237 0.580025068 1.487228085 2.614698261
## [131] 0.206662565 0.165110137 -0.713557580 -0.983434777 -0.693691169
## [136] 0.619494728 0.276063826 -0.483104007 0.251552914 -0.137485807
## [141] 0.896558099 -1.131402124 0.558283920 1.678193384 2.069400658
## [146] 0.888396815 -0.415100789 -0.660815187 0.748117966 -1.099365355
## [151] 0.377660091 -0.874150612 -0.606350362 -0.211369725 -0.847276755
## [156] 0.468932148 0.020681103 1.499110867 3.126409313 0.195770249
## [161] 2.117325973 1.817108123 1.150626089 0.115911690 2.512665732
## [166] 1.553432269 3.338992399 1.289444828 2.029379968 3.382984512
## [171] 2.457255539 1.684852166 0.535726173 0.403799194 -0.283028070
## [176] 0.444986829 -0.433816175 0.447145774 0.823288188 0.702182433
## [181] 0.827674987 0.861233509 1.451744630 0.349452445 0.850459079
## [186] 0.327858831 -0.702450476 -1.248961917 0.008528714 1.104098600
## [191] -1.238293635 0.328369561 0.161403931 1.205972092 2.472945399
## [196] 0.328072560 2.255412751 2.861535473 1.438188479 1.252317156
Berdasarkan plot time series tersebut terlihat bahwa data cenderung stasioner pada rataan.
Berdasarkan plot ACF tersebut terlihat bahwa plot ACF cenderung tails off dan cenderung membentuk pola grafik sinus.
Berdasarkan plot PACF tersebut, terlihat bahwa plot PACF cuts off pada lag kedua, sejalan dengan teori yang ada.
## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x x x x o o o o o
## 1 x o o o o o o o o o o o o o
## 2 x o o o o o o o o o o o o o
## 3 o x o o o o o o o o o o o o
## 4 o x o o o o o o o o o o o o
## 5 x x o o o o o o o o o o o o
## 6 x x x o x o o o o o o o o o
## 7 x x x x x o o o o o o o o o
Berdasarkan pola segitiga nol pada plot EACF, terlihat bahwa segitiga nol berada pada ordo AR(1) dan ordo MA(1).
## [1] 0.705141640 0.249970706 1.630575045 2.804006969 -0.632864527
## [6] -0.349744995 -0.845998100 -0.688856343 -1.343701238 -2.245002413
## [11] -1.467312487 -0.456788733 -1.309590275 -1.564831822 -0.839692440
## [16] -1.685225610 -1.976351619 -1.240349843 -0.300580252 -1.364600891
## [21] -0.280115127 -1.099028358 0.133222967 -0.472015853 -1.261331210
## [26] -1.010989626 -2.595626463 -2.539914335 0.135085568 -0.587623429
## [31] 0.062873138 -1.121668605 -0.196751533 -0.598041725 -0.725739020
## [36] -1.053296316 -1.663004703 -1.422636980 -0.629945133 -0.686128345
## [41] 0.217580950 -0.722747914 -0.160691574 0.101011878 -0.566744798
## [46] 0.073656410 1.243158084 2.470800226 0.711268152 -0.290944268
## [51] -0.374751162 -0.221706023 -0.364932764 0.703316452 1.867772149
## [56] 0.955796044 2.469579745 2.993735248 2.635760644 2.769842614
## [61] 1.551591476 1.846947073 -0.744491076 0.559766909 -0.306104106
## [66] -0.806377951 -0.469489115 0.150443138 -0.332375194 -0.888023314
## [71] 0.535932096 -0.318263458 -2.401184208 -1.487630879 -1.368579826
## [76] -0.989775631 1.356114931 1.709392162 -0.633670165 1.770677990
## [81] 0.274219021 1.000219974 -0.808515059 -0.871916643 -2.151063102
## [86] -0.350407053 -3.234663999 -2.279539723 -1.114053857 -2.319244900
## [91] -1.430136190 -2.322809507 -1.975066728 -2.531479937 -2.399113628
## [96] -0.538788396 -2.628538819 -1.931605552 -1.159526399 -1.014676079
## [101] -0.432185537 -0.549558850 1.264051868 1.430515867 2.598609523
## [106] 0.061890650 -0.617448193 0.276831567 0.101251347 -0.620719712
## [111] -0.443733826 0.087912564 0.726037551 -0.684766508 -1.100115503
## [116] -0.591801408 -1.364273590 -1.231732070 0.123787254 -0.817652540
## [121] 0.574641597 -1.117586460 0.410843111 2.937392529 0.659205141
## [126] 0.941254795 0.681051840 1.142075967 1.496952499 0.558230240
## [131] 0.639936785 1.367521449 1.786498994 1.017829854 -0.003569923
## [136] 0.746169264 0.679962460 1.432293504 1.069743017 -0.230637043
## [141] -0.706067505 -0.275492766 -1.710354850 -0.992195806 -0.425040791
## [146] -0.170944748 1.336619651 -0.579629920 -1.605344237 0.580025068
## [151] 1.487228085 2.614698261 0.206662565 0.165110137 -0.713557580
## [156] -0.983434777 -0.693691169 0.619494728 0.276063826 -0.483104007
## [161] 0.251552914 -0.137485807 0.896558099 -1.131402124 0.558283920
## [166] 1.678193384 2.069400658 0.888396815 -0.415100789 -0.660815187
## [171] 0.748117966 -1.099365355 0.377660091 -0.874150612 -0.606350362
## [176] -0.211369725 -0.847276755 0.468932148 0.020681103 1.499110867
## [181] 3.126409313 0.195770249 2.117325973 1.817108123 1.150626089
## [186] 0.115911690 2.512665732 1.553432269 3.338992399 1.289444828
## [191] 2.029379968 3.382984512 2.457255539 1.684852166 0.535726173
## [196] 0.403799194 -0.283028070 0.444986829 -0.433816175 0.447145774
## [201] 0.823288188 0.702182433 0.827674987 0.861233509 1.451744630
## [206] 0.349452445 0.850459079 0.327858831 -0.702450476 -1.248961917
## [211] 0.008528714 1.104098600 -1.238293635 0.328369561 0.161403931
## [216] 1.205972092 2.472945399 0.328072560 2.255412751 2.861535473
## [221] 1.438188479 1.252317156 -0.858881100 -0.016332851 -2.350702233
## [226] -1.059086108 -1.183245283 -0.729447937 -0.573160493 -0.584550104
## [231] -2.000657480 -2.597426055 -1.566350686 -0.911930377 -0.154070741
## [236] 0.761900065 0.611286982 -1.080733526 -0.934623033 -1.229103342
## [241] -1.519320967 1.469792754 -0.220051190 2.156993581 0.787710783
## [246] 0.837655525 2.535764561 1.600514779 0.836188700 1.122573472
## [251] 0.581180746 0.890620773 0.516259386 1.247959408 1.730458037
## [256] 2.313668714 -0.659434350 -1.363925887 -0.804814278 0.369415933
## [261] -0.049849610 -1.345918269 -0.945586014 -1.011022981 -0.637770952
## [266] -0.106931692 -1.447537800 -1.883412836 -1.528475651 0.067848989
## [271] -0.362758876 -0.281027342 -0.899903984 -0.837891287 -0.486910734
## [276] -0.172382511 -0.743789022 -2.054865504 -1.950659938 -3.411407367
## [281] -3.223933375 -2.625967645 -2.060706822 -1.556568946 -1.930909946
## [286] -3.625011258 -2.280016976 1.036139594 -1.106346059 -1.261263207
## [291] 1.334929741 -0.665011734 0.315274152 0.125254497 -0.676028950
## [296] -1.911686502 -3.376249052
## [1] 0.929297848 0.705141640 0.249970706 1.630575045 2.804006969
## [6] -0.632864527 -0.349744995 -0.845998100 -0.688856343 -1.343701238
## [11] -2.245002413 -1.467312487 -0.456788733 -1.309590275 -1.564831822
## [16] -0.839692440 -1.685225610 -1.976351619 -1.240349843 -0.300580252
## [21] -1.364600891 -0.280115127 -1.099028358 0.133222967 -0.472015853
## [26] -1.261331210 -1.010989626 -2.595626463 -2.539914335 0.135085568
## [31] -0.587623429 0.062873138 -1.121668605 -0.196751533 -0.598041725
## [36] -0.725739020 -1.053296316 -1.663004703 -1.422636980 -0.629945133
## [41] -0.686128345 0.217580950 -0.722747914 -0.160691574 0.101011878
## [46] -0.566744798 0.073656410 1.243158084 2.470800226 0.711268152
## [51] -0.290944268 -0.374751162 -0.221706023 -0.364932764 0.703316452
## [56] 1.867772149 0.955796044 2.469579745 2.993735248 2.635760644
## [61] 2.769842614 1.551591476 1.846947073 -0.744491076 0.559766909
## [66] -0.306104106 -0.806377951 -0.469489115 0.150443138 -0.332375194
## [71] -0.888023314 0.535932096 -0.318263458 -2.401184208 -1.487630879
## [76] -1.368579826 -0.989775631 1.356114931 1.709392162 -0.633670165
## [81] 1.770677990 0.274219021 1.000219974 -0.808515059 -0.871916643
## [86] -2.151063102 -0.350407053 -3.234663999 -2.279539723 -1.114053857
## [91] -2.319244900 -1.430136190 -2.322809507 -1.975066728 -2.531479937
## [96] -2.399113628 -0.538788396 -2.628538819 -1.931605552 -1.159526399
## [101] -1.014676079 -0.432185537 -0.549558850 1.264051868 1.430515867
## [106] 2.598609523 0.061890650 -0.617448193 0.276831567 0.101251347
## [111] -0.620719712 -0.443733826 0.087912564 0.726037551 -0.684766508
## [116] -1.100115503 -0.591801408 -1.364273590 -1.231732070 0.123787254
## [121] -0.817652540 0.574641597 -1.117586460 0.410843111 2.937392529
## [126] 0.659205141 0.941254795 0.681051840 1.142075967 1.496952499
## [131] 0.558230240 0.639936785 1.367521449 1.786498994 1.017829854
## [136] -0.003569923 0.746169264 0.679962460 1.432293504 1.069743017
## [141] -0.230637043 -0.706067505 -0.275492766 -1.710354850 -0.992195806
## [146] -0.425040791 -0.170944748 1.336619651 -0.579629920 -1.605344237
## [151] 0.580025068 1.487228085 2.614698261 0.206662565 0.165110137
## [156] -0.713557580 -0.983434777 -0.693691169 0.619494728 0.276063826
## [161] -0.483104007 0.251552914 -0.137485807 0.896558099 -1.131402124
## [166] 0.558283920 1.678193384 2.069400658 0.888396815 -0.415100789
## [171] -0.660815187 0.748117966 -1.099365355 0.377660091 -0.874150612
## [176] -0.606350362 -0.211369725 -0.847276755 0.468932148 0.020681103
## [181] 1.499110867 3.126409313 0.195770249 2.117325973 1.817108123
## [186] 1.150626089 0.115911690 2.512665732 1.553432269 3.338992399
## [191] 1.289444828 2.029379968 3.382984512 2.457255539 1.684852166
## [196] 0.535726173 0.403799194 -0.283028070 0.444986829 -0.433816175
## [201] 0.447145774 0.823288188 0.702182433 0.827674987 0.861233509
## [206] 1.451744630 0.349452445 0.850459079 0.327858831 -0.702450476
## [211] -1.248961917 0.008528714 1.104098600 -1.238293635 0.328369561
## [216] 0.161403931 1.205972092 2.472945399 0.328072560 2.255412751
## [221] 2.861535473 1.438188479 1.252317156 -0.858881100 -0.016332851
## [226] -2.350702233 -1.059086108 -1.183245283 -0.729447937 -0.573160493
## [231] -0.584550104 -2.000657480 -2.597426055 -1.566350686 -0.911930377
## [236] -0.154070741 0.761900065 0.611286982 -1.080733526 -0.934623033
## [241] -1.229103342 -1.519320967 1.469792754 -0.220051190 2.156993581
## [246] 0.787710783 0.837655525 2.535764561 1.600514779 0.836188700
## [251] 1.122573472 0.581180746 0.890620773 0.516259386 1.247959408
## [256] 1.730458037 2.313668714 -0.659434350 -1.363925887 -0.804814278
## [261] 0.369415933 -0.049849610 -1.345918269 -0.945586014 -1.011022981
## [266] -0.637770952 -0.106931692 -1.447537800 -1.883412836 -1.528475651
## [271] 0.067848989 -0.362758876 -0.281027342 -0.899903984 -0.837891287
## [276] -0.486910734 -0.172382511 -0.743789022 -2.054865504 -1.950659938
## [281] -3.411407367 -3.223933375 -2.625967645 -2.060706822 -1.556568946
## [286] -1.930909946 -3.625011258 -2.280016976 1.036139594 -1.106346059
## [291] -1.261263207 1.334929741 -0.665011734 0.315274152 0.125254497
## [296] -0.676028950 -1.911686502
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-1}\).
## [1] -0.897909868 0.929297848 0.705141640 0.249970706 1.630575045
## [6] 2.804006969 -0.632864527 -0.349744995 -0.845998100 -0.688856343
## [11] -1.343701238 -2.245002413 -1.467312487 -0.456788733 -1.309590275
## [16] -1.564831822 -0.839692440 -1.685225610 -1.976351619 -1.240349843
## [21] -0.300580252 -1.364600891 -0.280115127 -1.099028358 0.133222967
## [26] -0.472015853 -1.261331210 -1.010989626 -2.595626463 -2.539914335
## [31] 0.135085568 -0.587623429 0.062873138 -1.121668605 -0.196751533
## [36] -0.598041725 -0.725739020 -1.053296316 -1.663004703 -1.422636980
## [41] -0.629945133 -0.686128345 0.217580950 -0.722747914 -0.160691574
## [46] 0.101011878 -0.566744798 0.073656410 1.243158084 2.470800226
## [51] 0.711268152 -0.290944268 -0.374751162 -0.221706023 -0.364932764
## [56] 0.703316452 1.867772149 0.955796044 2.469579745 2.993735248
## [61] 2.635760644 2.769842614 1.551591476 1.846947073 -0.744491076
## [66] 0.559766909 -0.306104106 -0.806377951 -0.469489115 0.150443138
## [71] -0.332375194 -0.888023314 0.535932096 -0.318263458 -2.401184208
## [76] -1.487630879 -1.368579826 -0.989775631 1.356114931 1.709392162
## [81] -0.633670165 1.770677990 0.274219021 1.000219974 -0.808515059
## [86] -0.871916643 -2.151063102 -0.350407053 -3.234663999 -2.279539723
## [91] -1.114053857 -2.319244900 -1.430136190 -2.322809507 -1.975066728
## [96] -2.531479937 -2.399113628 -0.538788396 -2.628538819 -1.931605552
## [101] -1.159526399 -1.014676079 -0.432185537 -0.549558850 1.264051868
## [106] 1.430515867 2.598609523 0.061890650 -0.617448193 0.276831567
## [111] 0.101251347 -0.620719712 -0.443733826 0.087912564 0.726037551
## [116] -0.684766508 -1.100115503 -0.591801408 -1.364273590 -1.231732070
## [121] 0.123787254 -0.817652540 0.574641597 -1.117586460 0.410843111
## [126] 2.937392529 0.659205141 0.941254795 0.681051840 1.142075967
## [131] 1.496952499 0.558230240 0.639936785 1.367521449 1.786498994
## [136] 1.017829854 -0.003569923 0.746169264 0.679962460 1.432293504
## [141] 1.069743017 -0.230637043 -0.706067505 -0.275492766 -1.710354850
## [146] -0.992195806 -0.425040791 -0.170944748 1.336619651 -0.579629920
## [151] -1.605344237 0.580025068 1.487228085 2.614698261 0.206662565
## [156] 0.165110137 -0.713557580 -0.983434777 -0.693691169 0.619494728
## [161] 0.276063826 -0.483104007 0.251552914 -0.137485807 0.896558099
## [166] -1.131402124 0.558283920 1.678193384 2.069400658 0.888396815
## [171] -0.415100789 -0.660815187 0.748117966 -1.099365355 0.377660091
## [176] -0.874150612 -0.606350362 -0.211369725 -0.847276755 0.468932148
## [181] 0.020681103 1.499110867 3.126409313 0.195770249 2.117325973
## [186] 1.817108123 1.150626089 0.115911690 2.512665732 1.553432269
## [191] 3.338992399 1.289444828 2.029379968 3.382984512 2.457255539
## [196] 1.684852166 0.535726173 0.403799194 -0.283028070 0.444986829
## [201] -0.433816175 0.447145774 0.823288188 0.702182433 0.827674987
## [206] 0.861233509 1.451744630 0.349452445 0.850459079 0.327858831
## [211] -0.702450476 -1.248961917 0.008528714 1.104098600 -1.238293635
## [216] 0.328369561 0.161403931 1.205972092 2.472945399 0.328072560
## [221] 2.255412751 2.861535473 1.438188479 1.252317156 -0.858881100
## [226] -0.016332851 -2.350702233 -1.059086108 -1.183245283 -0.729447937
## [231] -0.573160493 -0.584550104 -2.000657480 -2.597426055 -1.566350686
## [236] -0.911930377 -0.154070741 0.761900065 0.611286982 -1.080733526
## [241] -0.934623033 -1.229103342 -1.519320967 1.469792754 -0.220051190
## [246] 2.156993581 0.787710783 0.837655525 2.535764561 1.600514779
## [251] 0.836188700 1.122573472 0.581180746 0.890620773 0.516259386
## [256] 1.247959408 1.730458037 2.313668714 -0.659434350 -1.363925887
## [261] -0.804814278 0.369415933 -0.049849610 -1.345918269 -0.945586014
## [266] -1.011022981 -0.637770952 -0.106931692 -1.447537800 -1.883412836
## [271] -1.528475651 0.067848989 -0.362758876 -0.281027342 -0.899903984
## [276] -0.837891287 -0.486910734 -0.172382511 -0.743789022 -2.054865504
## [281] -1.950659938 -3.411407367 -3.223933375 -2.625967645 -2.060706822
## [286] -1.556568946 -1.930909946 -3.625011258 -2.280016976 1.036139594
## [291] -1.106346059 -1.261263207 1.334929741 -0.665011734 0.315274152
## [296] 0.125254497 -0.676028950
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-2}\).
## [1] 0.131540851 -0.897909868 0.929297848 0.705141640 0.249970706
## [6] 1.630575045 2.804006969 -0.632864527 -0.349744995 -0.845998100
## [11] -0.688856343 -1.343701238 -2.245002413 -1.467312487 -0.456788733
## [16] -1.309590275 -1.564831822 -0.839692440 -1.685225610 -1.976351619
## [21] -1.240349843 -0.300580252 -1.364600891 -0.280115127 -1.099028358
## [26] 0.133222967 -0.472015853 -1.261331210 -1.010989626 -2.595626463
## [31] -2.539914335 0.135085568 -0.587623429 0.062873138 -1.121668605
## [36] -0.196751533 -0.598041725 -0.725739020 -1.053296316 -1.663004703
## [41] -1.422636980 -0.629945133 -0.686128345 0.217580950 -0.722747914
## [46] -0.160691574 0.101011878 -0.566744798 0.073656410 1.243158084
## [51] 2.470800226 0.711268152 -0.290944268 -0.374751162 -0.221706023
## [56] -0.364932764 0.703316452 1.867772149 0.955796044 2.469579745
## [61] 2.993735248 2.635760644 2.769842614 1.551591476 1.846947073
## [66] -0.744491076 0.559766909 -0.306104106 -0.806377951 -0.469489115
## [71] 0.150443138 -0.332375194 -0.888023314 0.535932096 -0.318263458
## [76] -2.401184208 -1.487630879 -1.368579826 -0.989775631 1.356114931
## [81] 1.709392162 -0.633670165 1.770677990 0.274219021 1.000219974
## [86] -0.808515059 -0.871916643 -2.151063102 -0.350407053 -3.234663999
## [91] -2.279539723 -1.114053857 -2.319244900 -1.430136190 -2.322809507
## [96] -1.975066728 -2.531479937 -2.399113628 -0.538788396 -2.628538819
## [101] -1.931605552 -1.159526399 -1.014676079 -0.432185537 -0.549558850
## [106] 1.264051868 1.430515867 2.598609523 0.061890650 -0.617448193
## [111] 0.276831567 0.101251347 -0.620719712 -0.443733826 0.087912564
## [116] 0.726037551 -0.684766508 -1.100115503 -0.591801408 -1.364273590
## [121] -1.231732070 0.123787254 -0.817652540 0.574641597 -1.117586460
## [126] 0.410843111 2.937392529 0.659205141 0.941254795 0.681051840
## [131] 1.142075967 1.496952499 0.558230240 0.639936785 1.367521449
## [136] 1.786498994 1.017829854 -0.003569923 0.746169264 0.679962460
## [141] 1.432293504 1.069743017 -0.230637043 -0.706067505 -0.275492766
## [146] -1.710354850 -0.992195806 -0.425040791 -0.170944748 1.336619651
## [151] -0.579629920 -1.605344237 0.580025068 1.487228085 2.614698261
## [156] 0.206662565 0.165110137 -0.713557580 -0.983434777 -0.693691169
## [161] 0.619494728 0.276063826 -0.483104007 0.251552914 -0.137485807
## [166] 0.896558099 -1.131402124 0.558283920 1.678193384 2.069400658
## [171] 0.888396815 -0.415100789 -0.660815187 0.748117966 -1.099365355
## [176] 0.377660091 -0.874150612 -0.606350362 -0.211369725 -0.847276755
## [181] 0.468932148 0.020681103 1.499110867 3.126409313 0.195770249
## [186] 2.117325973 1.817108123 1.150626089 0.115911690 2.512665732
## [191] 1.553432269 3.338992399 1.289444828 2.029379968 3.382984512
## [196] 2.457255539 1.684852166 0.535726173 0.403799194 -0.283028070
## [201] 0.444986829 -0.433816175 0.447145774 0.823288188 0.702182433
## [206] 0.827674987 0.861233509 1.451744630 0.349452445 0.850459079
## [211] 0.327858831 -0.702450476 -1.248961917 0.008528714 1.104098600
## [216] -1.238293635 0.328369561 0.161403931 1.205972092 2.472945399
## [221] 0.328072560 2.255412751 2.861535473 1.438188479 1.252317156
## [226] -0.858881100 -0.016332851 -2.350702233 -1.059086108 -1.183245283
## [231] -0.729447937 -0.573160493 -0.584550104 -2.000657480 -2.597426055
## [236] -1.566350686 -0.911930377 -0.154070741 0.761900065 0.611286982
## [241] -1.080733526 -0.934623033 -1.229103342 -1.519320967 1.469792754
## [246] -0.220051190 2.156993581 0.787710783 0.837655525 2.535764561
## [251] 1.600514779 0.836188700 1.122573472 0.581180746 0.890620773
## [256] 0.516259386 1.247959408 1.730458037 2.313668714 -0.659434350
## [261] -1.363925887 -0.804814278 0.369415933 -0.049849610 -1.345918269
## [266] -0.945586014 -1.011022981 -0.637770952 -0.106931692 -1.447537800
## [271] -1.883412836 -1.528475651 0.067848989 -0.362758876 -0.281027342
## [276] -0.899903984 -0.837891287 -0.486910734 -0.172382511 -0.743789022
## [281] -2.054865504 -1.950659938 -3.411407367 -3.223933375 -2.625967645
## [286] -2.060706822 -1.556568946 -1.930909946 -3.625011258 -2.280016976
## [291] 1.036139594 -1.106346059 -1.261263207 1.334929741 -0.665011734
## [296] 0.315274152 0.125254497
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-3}\).
## [1] 0.6334386
Nilai korelasi antara \(Y_t\) dan \(Y_{t-1}\) dari hasil simulasi mendekati perhitungan teoritis yaitu:
\[ \rho_{1} = \frac{\phi_1}{1-\phi_2} = \frac{0.5}{1 - 0.2} = 0.625 \]
## [1] 0.513792
Nilai korelasi antara \(Y_t\) dan \(Y_{t-2}\) dari hasil simulasi mendekati perhitungan teoritis yaitu:
\[ \rho_{2} = \frac{\phi_2 (1-\phi_2) + (\phi_1)^2}{1-\phi_2} = \frac{0.2(1-0.2) + (0.5)^2}{1 - 0.2} \approx 0.512 \]
## [1] 0.4262701
Nilai korelasi antara \(Y_t\) dan \(Y_{t-3}\) dari hasil simulasi mendekati perhitungan teoritis yaitu:
\[ \rho_{3} = \phi_1\rho_{3-1} + \phi_2\rho_{3-2} = \phi_1\rho_{2} + \phi_2\rho_{1} = 0.5(0.512) + 0.2(0.625) \approx 0.381 \]
Kesimpulan: Berdasarkan hasil analisis, model AR(2) merupakan model yang paling sesuai karena pola ACF dan PACF sudah konsisten dengan karakteristik teoritis AR(2), serta nilai autokorelasi empiris mendekati nilai teoritis. Meskipun hasil EACF menunjukkan kecenderungan ke ARMA(1,1), hal tersebut kemungkinan dipengaruhi oleh keterbatasan sampel atau kemiripan pola antar model. Dengan demikian, AR(2) tetap dapat dianggap sebagai representasi yang tepat untuk proses data bangkitan.
Rumus ARMA (2,2): \[ Y_t = \phi_1Y_{t-1}+\phi_2Y_{t-2}+e_t-\theta_1e_{t-1}-\theta_2e_{t-2} \] dengan \(\theta_1\) = 0.4, \(\theta_2\) = 0.6, \(\phi_1\) = 0.5, dan \(\phi_2\) = 0.2.
# Manual
set.seed(1010)
y.arma <- wn[1:2]
for (i in 3:n) {
y.arma[i] <- (0.5 * y.arma[i-1]) +
(0.2 * y.arma[i-2]) +
wn[i] +
(0.4 * wn[i-1]) +
(0.6 * wn[i-2])
}
y.arma## [1] 0.1315408507 -0.8979098678 0.6490584112 0.5670538453 1.1040755646
## [6] 2.1666708551 3.6156245817 1.4743903819 1.0850479575 -1.3613861046
## [11] -1.2338813084 -2.1243862581 -3.1939242710 -3.1701066986 -2.3896269389
## [16] -2.3718636432 -2.3621087157 -2.2508971826 -2.9596341124 -3.1539771138
## [21] -3.0418122366 -1.9823683078 -2.2289187469 -1.0062089885 -2.0297627901
## [26] -0.4744024462 -1.0781017468 -1.3701718025 -1.7987072509 -3.7568024608
## [31] -4.1847445327 -2.4382452466 -2.0575295717 -0.0911186175 -1.4490886232
## [36] -0.6076914446 -1.3497407211 -1.0830045097 -1.7024153434 -2.5197654093
## [41] -2.7198157113 -2.1968020303 -1.7916880403 -0.4348370521 -1.0473922237
## [46] -0.3192419282 -0.3969133154 -0.6227548509 -0.0924342754 0.9325738506
## [51] 3.0122573682 2.4454831403 1.4760431644 -0.0643679508 -0.5461730279
## [56] -0.6784658549 0.4243197444 1.9301390808 2.1248947815 3.9725614580
## [61] 4.5550447761 5.3150025930 5.6203880226 4.2409849098 4.1294892331
## [66] 0.9252426396 1.3701387231 -0.5288919875 -0.5929594477 -0.9757027592
## [71] -0.5211792791 -0.5538914081 -0.9307075093 -0.0187023456 -0.6367046077
## [76] -2.2069303334 -2.6390626365 -3.4043427022 -2.4297860883 0.1390567834
## [81] 1.6579727565 0.8637556592 2.5428452214 0.6022881181 2.1723143762
## [86] -0.2438956572 -0.5951906828 -2.9849387954 -1.7339822802 -4.6654646819
## [91] -3.7836495546 -3.9666681459 -4.1325902764 -3.0262664639 -4.2864109226
## [96] -3.7622722444 -4.7151923322 -4.5967456399 -3.0173218097 -4.2835223541
## [101] -3.3062941169 -3.5092919106 -2.6374499698 -1.5337718084 -1.3312387130
## [106] 0.7849170050 1.6064013034 3.9292469905 1.9596439795 0.9664737813
## [111] 0.0669866797 -0.1584849418 -0.4141202333 -0.6312709023 -0.4620127937
## [116] 0.4949622808 -0.3416039498 -0.9383995764 -1.4427075147 -2.2610634558
## [121] -2.1325223511 -1.1874697279 -1.5071768803 0.3218529332 -1.3783213458
## [126] 0.3085934849 2.4311778976 2.0806680196 2.9673723690 1.4530768428
## [131] 1.9792495800 2.3624139896 1.8422568201 1.7614003802 1.9584343073
## [136] 2.7174696443 2.5529423211 1.4754614153 1.3554392078 0.9762882119
## [141] 2.1519800469 2.0506378942 1.0566362664 -0.1564765121 -0.6963019933
## [146] -2.2441924588 -1.8416334056 -1.8481320229 -0.9362785485 1.0132172775
## [151] -0.1475489089 -1.0352244145 -0.4098905792 0.7560315697 3.5576045354
## [156] 2.1448787201 1.8165941197 -0.5235159857 -1.1697917262 -1.5151996272
## [161] -0.2480426055 0.1076470164 -0.0009816395 0.2239496072 -0.3267270455
## [166] 0.9924955244 -0.8552703690 0.6436579300 1.2226656780 3.0756483638
## [171] 2.7230731085 1.1818983317 -0.2938174133 0.2347314183 -1.1966072807
## [176] 0.3867847283 -1.3827057885 -0.7294145525 -0.9784002367 -1.2956348622
## [181] 0.0031996111 -0.3001120910 1.7887425967 3.7384623212 2.3458004938
## [186] 4.0714796600 2.7815006616 3.1478649220 1.6664269995 3.2494060618
## [191] 2.6280455760 5.4679647463 3.5571011493 4.5485533388 4.9684033961
## [196] 5.0280773241 4.6975450886 2.6840203628 1.6290009627 0.1999273113
## [201] 0.5740551176 -0.4256382849 0.5406114019 0.7418567933 1.2997851726
## [206] 1.6025208730 1.6136129632 2.2928430253 1.4468904019 1.8612868351
## [211] 0.8777139291 -0.0610314965 -1.3332268089 -0.9125263389 0.3581329351
## [216] -0.7915369668 0.4955112665 -0.4502244253 1.4675554008 3.0521765941
## [221] 2.0408339742 3.8704090143 3.9605441093 3.9360503187 3.5445138311
## [226] 0.5049588491 0.3915050022 -2.8725640338 -2.0091667120 -3.0173010664
## [231] -1.8381977156 -1.5748868384 -1.2514830640 -2.5783738178 -3.7484191100
## [236] -3.8057155958 -3.0969262848 -1.4586533030 0.1531135428 0.8236045641
## [241] -0.3790786936 -1.0001442542 -2.2513926710 -2.5717361239 0.1246023618
## [246] -0.5437266688 2.9508487574 1.5184775014 2.4469359872 3.3434532407
## [251] 3.1174139183 2.9978533478 2.4173578196 1.5319233546 1.7966371548
## [256] 1.2212161425 1.9888356266 2.5393974322 3.7546275738 1.3043079574
## [261] -0.2394983994 -1.7460452432 -0.7708653105 -0.3849718038 -1.1442085530
## [266] -1.5138630876 -2.1968083475 -1.6095317523 -0.9686538607 -1.8729730474
## [271] -2.5265869712 -3.1503634658 -1.6735889734 -1.2527046715 -0.3854214991
## [276] -1.2299702469 -1.3664692859 -1.3620096392 -0.8698815763 -1.1048884660
## [281] -2.4558106193 -3.2188795522 -5.4245906442 -5.7588922841 -5.9623854154
## [286] -5.0454539055 -3.9564322618 -3.7899616180 -5.3313166036 -4.8885674466
## [291] -2.0508739512 -2.0599004072 -1.0821178750 0.1666168226 -0.8877977615
## [296] 0.8502273032 -0.1476428826 -0.4367626603 -2.1069453839 -4.5465410226
# Dengan fungsi arima.sim
arma22 <- arima.sim(
list(order=c(2,0,2),
ar = c(0.5, 0.2),
ma = c(0.4, 0.6)),
n = 300
)
arma22## Time Series:
## Start = 1
## End = 300
## Frequency = 1
## [1] -1.3696174977 -1.7982846779 -3.7564803134 -4.1844989443 -2.4380580230
## [6] -2.0573868423 -0.0910098080 -1.4490056726 -0.6076282073 -1.3496925124
## [11] -1.0829677579 -1.7023873257 -2.5197440501 -2.7197994281 -2.1967896169
## [16] -1.7916785770 -0.4348298378 -1.0473867239 -0.3192377354 -0.3969101190
## [21] -0.6227524142 -0.0924324177 0.9325752668 3.0122584478 2.4454839633
## [26] 1.4760437919 -0.0643674725 -0.5461726632 -0.6784655769 0.4243199563
## [31] 1.9301392423 2.1248949046 3.9725615519 4.5550448477 5.3150026476
## [36] 5.6203880642 4.2409849415 4.1294892573 0.9252426580 1.3701387371
## [41] -0.5288919768 -0.5929594395 -0.9757027529 -0.5211792743 -0.5538914045
## [46] -0.9307075065 -0.0187023435 -0.6367046061 -2.2069303322 -2.6390626356
## [51] -3.4043427015 -2.4297860877 0.1390567838 1.6579727568 0.8637556594
## [56] 2.5428452215 0.6022881183 2.1723143763 -0.2438956571 -0.5951906827
## [61] -2.9849387954 -1.7339822802 -4.6654646818 -3.7836495546 -3.9666681459
## [66] -4.1325902763 -3.0262664639 -4.2864109226 -3.7622722444 -4.7151923322
## [71] -4.5967456399 -3.0173218097 -4.2835223541 -3.3062941169 -3.5092919106
## [76] -2.6374499698 -1.5337718084 -1.3312387130 0.7849170050 1.6064013034
## [81] 3.9292469905 1.9596439795 0.9664737813 0.0669866797 -0.1584849418
## [86] -0.4141202333 -0.6312709023 -0.4620127937 0.4949622808 -0.3416039498
## [91] -0.9383995764 -1.4427075147 -2.2610634558 -2.1325223511 -1.1874697279
## [96] -1.5071768803 0.3218529332 -1.3783213458 0.3085934849 2.4311778976
## [101] 2.0806680196 2.9673723690 1.4530768428 1.9792495800 2.3624139896
## [106] 1.8422568201 1.7614003802 1.9584343073 2.7174696443 2.5529423211
## [111] 1.4754614153 1.3554392078 0.9762882119 2.1519800469 2.0506378942
## [116] 1.0566362664 -0.1564765121 -0.6963019933 -2.2441924588 -1.8416334056
## [121] -1.8481320229 -0.9362785485 1.0132172775 -0.1475489089 -1.0352244145
## [126] -0.4098905792 0.7560315697 3.5576045354 2.1448787201 1.8165941197
## [131] -0.5235159857 -1.1697917262 -1.5151996272 -0.2480426055 0.1076470164
## [136] -0.0009816395 0.2239496072 -0.3267270455 0.9924955244 -0.8552703690
## [141] 0.6436579300 1.2226656780 3.0756483638 2.7230731085 1.1818983317
## [146] -0.2938174133 0.2347314183 -1.1966072807 0.3867847283 -1.3827057885
## [151] -0.7294145525 -0.9784002367 -1.2956348622 0.0031996111 -0.3001120910
## [156] 1.7887425967 3.7384623212 2.3458004938 4.0714796600 2.7815006616
## [161] 3.1478649220 1.6664269995 3.2494060618 2.6280455760 5.4679647463
## [166] 3.5571011493 4.5485533388 4.9684033961 5.0280773241 4.6975450886
## [171] 2.6840203628 1.6290009627 0.1999273113 0.5740551176 -0.4256382849
## [176] 0.5406114019 0.7418567933 1.2997851726 1.6025208730 1.6136129632
## [181] 2.2928430253 1.4468904019 1.8612868351 0.8777139291 -0.0610314965
## [186] -1.3332268089 -0.9125263389 0.3581329351 -0.7915369668 0.4955112665
## [191] -0.4502244253 1.4675554008 3.0521765941 2.0408339742 3.8704090143
## [196] 3.9605441093 3.9360503187 3.5445138311 0.5049588491 0.3915050022
## [201] -2.8725640338 -2.0091667120 -3.0173010664 -1.8381977156 -1.5748868384
## [206] -1.2514830640 -2.5783738178 -3.7484191100 -3.8057155958 -3.0969262848
## [211] -1.4586533030 0.1531135428 0.8236045641 -0.3790786936 -1.0001442542
## [216] -2.2513926710 -2.5717361239 0.1246023618 -0.5437266688 2.9508487574
## [221] 1.5184775014 2.4469359872 3.3434532407 3.1174139183 2.9978533478
## [226] 2.4173578196 1.5319233546 1.7966371548 1.2212161425 1.9888356266
## [231] 2.5393974322 3.7546275738 1.3043079574 -0.2394983994 -1.7460452432
## [236] -0.7708653105 -0.3849718038 -1.1442085530 -1.5138630876 -2.1968083475
## [241] -1.6095317523 -0.9686538607 -1.8729730474 -2.5265869712 -3.1503634658
## [246] -1.6735889734 -1.2527046715 -0.3854214991 -1.2299702469 -1.3664692859
## [251] -1.3620096392 -0.8698815763 -1.1048884660 -2.4558106193 -3.2188795522
## [256] -5.4245906442 -5.7588922841 -5.9623854154 -5.0454539055 -3.9564322618
## [261] -3.7899616180 -5.3313166036 -4.8885674466 -2.0508739512 -2.0599004072
## [266] -1.0821178750 0.1666168226 -0.8877977615 0.8502273032 -0.1476428826
## [271] -0.4367626603 -2.1069453839 -4.5465410226 -4.8087456252 -4.2289076874
## [276] -1.9178765546 -2.2804864851 -0.4098672855 -1.6178574433 -0.0137269804
## [281] 0.0122011943 1.4282807605 0.6560697941 0.7567608782 -0.8228605730
## [286] -1.1860435575 -1.8041134735 -0.4466562647 -0.3954967938 2.1487643901
## [291] 0.9586140341 2.2653125680 0.2588626762 1.0329576284 -0.9297517508
## [296] -1.2122608974 -2.3845725749 -5.0393956847 -3.3418821354 -4.8054361842
Berdasarkan plot time series tersebut, terlihat bahwa model ARMA(2,2) cenderung stasioner dalam rataan.
Berdasarkan plot ACF tersebut, terlihat bahwa model ARMA(2,2) hasil simulasi memiliki plot ACF yang tails off, sesuai dengan teori yang ada.
Berdasarkan plot PACF tersebut, terlihat bahwa model ARMA(2,2) hasil simulasi memiliki plot PACF yang tails off, sesuai dengan teori yang ada.
## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x x x x x o o o o
## 1 o x o o o o o o o o o o o o
## 2 x x x o o o o o o o o o o o
## 3 x x o o o o o o o o o o o o
## 4 x x x o o o o o o o o o o o
## 5 x x x o o o o o o o o o o o
## 6 x x x o o o o o o o o o o o
## 7 x x x o o o o o o o o o o o
## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x x x x x x x o o
## 1 o x o o o o o o o o o o o o
## 2 x x x o o o o o o o o o o o
## 3 x x x o o o o o o o o o o o
## 4 x x o o o o o o o o o o o o
## 5 x x x o x o o o o o o o o o
## 6 o x o o x o o o o o o o o o
## 7 x x x o x x o o o o o o o o
Berdasarkan pola segitiga nol pada plot EACF, terlihat bahwa segitiga nol berada pada ordo AR(1) dan ordo MA(2).
## [1] -4.1844989443 -2.4380580230 -2.0573868423 -0.0910098080 -1.4490056726
## [6] -0.6076282073 -1.3496925124 -1.0829677579 -1.7023873257 -2.5197440501
## [11] -2.7197994281 -2.1967896169 -1.7916785770 -0.4348298378 -1.0473867239
## [16] -0.3192377354 -0.3969101190 -0.6227524142 -0.0924324177 0.9325752668
## [21] 3.0122584478 2.4454839633 1.4760437919 -0.0643674725 -0.5461726632
## [26] -0.6784655769 0.4243199563 1.9301392423 2.1248949046 3.9725615519
## [31] 4.5550448477 5.3150026476 5.6203880642 4.2409849415 4.1294892573
## [36] 0.9252426580 1.3701387371 -0.5288919768 -0.5929594395 -0.9757027529
## [41] -0.5211792743 -0.5538914045 -0.9307075065 -0.0187023435 -0.6367046061
## [46] -2.2069303322 -2.6390626356 -3.4043427015 -2.4297860877 0.1390567838
## [51] 1.6579727568 0.8637556594 2.5428452215 0.6022881183 2.1723143763
## [56] -0.2438956571 -0.5951906827 -2.9849387954 -1.7339822802 -4.6654646818
## [61] -3.7836495546 -3.9666681459 -4.1325902763 -3.0262664639 -4.2864109226
## [66] -3.7622722444 -4.7151923322 -4.5967456399 -3.0173218097 -4.2835223541
## [71] -3.3062941169 -3.5092919106 -2.6374499698 -1.5337718084 -1.3312387130
## [76] 0.7849170050 1.6064013034 3.9292469905 1.9596439795 0.9664737813
## [81] 0.0669866797 -0.1584849418 -0.4141202333 -0.6312709023 -0.4620127937
## [86] 0.4949622808 -0.3416039498 -0.9383995764 -1.4427075147 -2.2610634558
## [91] -2.1325223511 -1.1874697279 -1.5071768803 0.3218529332 -1.3783213458
## [96] 0.3085934849 2.4311778976 2.0806680196 2.9673723690 1.4530768428
## [101] 1.9792495800 2.3624139896 1.8422568201 1.7614003802 1.9584343073
## [106] 2.7174696443 2.5529423211 1.4754614153 1.3554392078 0.9762882119
## [111] 2.1519800469 2.0506378942 1.0566362664 -0.1564765121 -0.6963019933
## [116] -2.2441924588 -1.8416334056 -1.8481320229 -0.9362785485 1.0132172775
## [121] -0.1475489089 -1.0352244145 -0.4098905792 0.7560315697 3.5576045354
## [126] 2.1448787201 1.8165941197 -0.5235159857 -1.1697917262 -1.5151996272
## [131] -0.2480426055 0.1076470164 -0.0009816395 0.2239496072 -0.3267270455
## [136] 0.9924955244 -0.8552703690 0.6436579300 1.2226656780 3.0756483638
## [141] 2.7230731085 1.1818983317 -0.2938174133 0.2347314183 -1.1966072807
## [146] 0.3867847283 -1.3827057885 -0.7294145525 -0.9784002367 -1.2956348622
## [151] 0.0031996111 -0.3001120910 1.7887425967 3.7384623212 2.3458004938
## [156] 4.0714796600 2.7815006616 3.1478649220 1.6664269995 3.2494060618
## [161] 2.6280455760 5.4679647463 3.5571011493 4.5485533388 4.9684033961
## [166] 5.0280773241 4.6975450886 2.6840203628 1.6290009627 0.1999273113
## [171] 0.5740551176 -0.4256382849 0.5406114019 0.7418567933 1.2997851726
## [176] 1.6025208730 1.6136129632 2.2928430253 1.4468904019 1.8612868351
## [181] 0.8777139291 -0.0610314965 -1.3332268089 -0.9125263389 0.3581329351
## [186] -0.7915369668 0.4955112665 -0.4502244253 1.4675554008 3.0521765941
## [191] 2.0408339742 3.8704090143 3.9605441093 3.9360503187 3.5445138311
## [196] 0.5049588491 0.3915050022 -2.8725640338 -2.0091667120 -3.0173010664
## [201] -1.8381977156 -1.5748868384 -1.2514830640 -2.5783738178 -3.7484191100
## [206] -3.8057155958 -3.0969262848 -1.4586533030 0.1531135428 0.8236045641
## [211] -0.3790786936 -1.0001442542 -2.2513926710 -2.5717361239 0.1246023618
## [216] -0.5437266688 2.9508487574 1.5184775014 2.4469359872 3.3434532407
## [221] 3.1174139183 2.9978533478 2.4173578196 1.5319233546 1.7966371548
## [226] 1.2212161425 1.9888356266 2.5393974322 3.7546275738 1.3043079574
## [231] -0.2394983994 -1.7460452432 -0.7708653105 -0.3849718038 -1.1442085530
## [236] -1.5138630876 -2.1968083475 -1.6095317523 -0.9686538607 -1.8729730474
## [241] -2.5265869712 -3.1503634658 -1.6735889734 -1.2527046715 -0.3854214991
## [246] -1.2299702469 -1.3664692859 -1.3620096392 -0.8698815763 -1.1048884660
## [251] -2.4558106193 -3.2188795522 -5.4245906442 -5.7588922841 -5.9623854154
## [256] -5.0454539055 -3.9564322618 -3.7899616180 -5.3313166036 -4.8885674466
## [261] -2.0508739512 -2.0599004072 -1.0821178750 0.1666168226 -0.8877977615
## [266] 0.8502273032 -0.1476428826 -0.4367626603 -2.1069453839 -4.5465410226
## [271] -4.8087456252 -4.2289076874 -1.9178765546 -2.2804864851 -0.4098672855
## [276] -1.6178574433 -0.0137269804 0.0122011943 1.4282807605 0.6560697941
## [281] 0.7567608782 -0.8228605730 -1.1860435575 -1.8041134735 -0.4466562647
## [286] -0.3954967938 2.1487643901 0.9586140341 2.2653125680 0.2588626762
## [291] 1.0329576284 -0.9297517508 -1.2122608974 -2.3845725749 -5.0393956847
## [296] -3.3418821354 -4.8054361842
## [1] -3.7564803134 -4.1844989443 -2.4380580230 -2.0573868423 -0.0910098080
## [6] -1.4490056726 -0.6076282073 -1.3496925124 -1.0829677579 -1.7023873257
## [11] -2.5197440501 -2.7197994281 -2.1967896169 -1.7916785770 -0.4348298378
## [16] -1.0473867239 -0.3192377354 -0.3969101190 -0.6227524142 -0.0924324177
## [21] 0.9325752668 3.0122584478 2.4454839633 1.4760437919 -0.0643674725
## [26] -0.5461726632 -0.6784655769 0.4243199563 1.9301392423 2.1248949046
## [31] 3.9725615519 4.5550448477 5.3150026476 5.6203880642 4.2409849415
## [36] 4.1294892573 0.9252426580 1.3701387371 -0.5288919768 -0.5929594395
## [41] -0.9757027529 -0.5211792743 -0.5538914045 -0.9307075065 -0.0187023435
## [46] -0.6367046061 -2.2069303322 -2.6390626356 -3.4043427015 -2.4297860877
## [51] 0.1390567838 1.6579727568 0.8637556594 2.5428452215 0.6022881183
## [56] 2.1723143763 -0.2438956571 -0.5951906827 -2.9849387954 -1.7339822802
## [61] -4.6654646818 -3.7836495546 -3.9666681459 -4.1325902763 -3.0262664639
## [66] -4.2864109226 -3.7622722444 -4.7151923322 -4.5967456399 -3.0173218097
## [71] -4.2835223541 -3.3062941169 -3.5092919106 -2.6374499698 -1.5337718084
## [76] -1.3312387130 0.7849170050 1.6064013034 3.9292469905 1.9596439795
## [81] 0.9664737813 0.0669866797 -0.1584849418 -0.4141202333 -0.6312709023
## [86] -0.4620127937 0.4949622808 -0.3416039498 -0.9383995764 -1.4427075147
## [91] -2.2610634558 -2.1325223511 -1.1874697279 -1.5071768803 0.3218529332
## [96] -1.3783213458 0.3085934849 2.4311778976 2.0806680196 2.9673723690
## [101] 1.4530768428 1.9792495800 2.3624139896 1.8422568201 1.7614003802
## [106] 1.9584343073 2.7174696443 2.5529423211 1.4754614153 1.3554392078
## [111] 0.9762882119 2.1519800469 2.0506378942 1.0566362664 -0.1564765121
## [116] -0.6963019933 -2.2441924588 -1.8416334056 -1.8481320229 -0.9362785485
## [121] 1.0132172775 -0.1475489089 -1.0352244145 -0.4098905792 0.7560315697
## [126] 3.5576045354 2.1448787201 1.8165941197 -0.5235159857 -1.1697917262
## [131] -1.5151996272 -0.2480426055 0.1076470164 -0.0009816395 0.2239496072
## [136] -0.3267270455 0.9924955244 -0.8552703690 0.6436579300 1.2226656780
## [141] 3.0756483638 2.7230731085 1.1818983317 -0.2938174133 0.2347314183
## [146] -1.1966072807 0.3867847283 -1.3827057885 -0.7294145525 -0.9784002367
## [151] -1.2956348622 0.0031996111 -0.3001120910 1.7887425967 3.7384623212
## [156] 2.3458004938 4.0714796600 2.7815006616 3.1478649220 1.6664269995
## [161] 3.2494060618 2.6280455760 5.4679647463 3.5571011493 4.5485533388
## [166] 4.9684033961 5.0280773241 4.6975450886 2.6840203628 1.6290009627
## [171] 0.1999273113 0.5740551176 -0.4256382849 0.5406114019 0.7418567933
## [176] 1.2997851726 1.6025208730 1.6136129632 2.2928430253 1.4468904019
## [181] 1.8612868351 0.8777139291 -0.0610314965 -1.3332268089 -0.9125263389
## [186] 0.3581329351 -0.7915369668 0.4955112665 -0.4502244253 1.4675554008
## [191] 3.0521765941 2.0408339742 3.8704090143 3.9605441093 3.9360503187
## [196] 3.5445138311 0.5049588491 0.3915050022 -2.8725640338 -2.0091667120
## [201] -3.0173010664 -1.8381977156 -1.5748868384 -1.2514830640 -2.5783738178
## [206] -3.7484191100 -3.8057155958 -3.0969262848 -1.4586533030 0.1531135428
## [211] 0.8236045641 -0.3790786936 -1.0001442542 -2.2513926710 -2.5717361239
## [216] 0.1246023618 -0.5437266688 2.9508487574 1.5184775014 2.4469359872
## [221] 3.3434532407 3.1174139183 2.9978533478 2.4173578196 1.5319233546
## [226] 1.7966371548 1.2212161425 1.9888356266 2.5393974322 3.7546275738
## [231] 1.3043079574 -0.2394983994 -1.7460452432 -0.7708653105 -0.3849718038
## [236] -1.1442085530 -1.5138630876 -2.1968083475 -1.6095317523 -0.9686538607
## [241] -1.8729730474 -2.5265869712 -3.1503634658 -1.6735889734 -1.2527046715
## [246] -0.3854214991 -1.2299702469 -1.3664692859 -1.3620096392 -0.8698815763
## [251] -1.1048884660 -2.4558106193 -3.2188795522 -5.4245906442 -5.7588922841
## [256] -5.9623854154 -5.0454539055 -3.9564322618 -3.7899616180 -5.3313166036
## [261] -4.8885674466 -2.0508739512 -2.0599004072 -1.0821178750 0.1666168226
## [266] -0.8877977615 0.8502273032 -0.1476428826 -0.4367626603 -2.1069453839
## [271] -4.5465410226 -4.8087456252 -4.2289076874 -1.9178765546 -2.2804864851
## [276] -0.4098672855 -1.6178574433 -0.0137269804 0.0122011943 1.4282807605
## [281] 0.6560697941 0.7567608782 -0.8228605730 -1.1860435575 -1.8041134735
## [286] -0.4466562647 -0.3954967938 2.1487643901 0.9586140341 2.2653125680
## [291] 0.2588626762 1.0329576284 -0.9297517508 -1.2122608974 -2.3845725749
## [296] -5.0393956847 -3.3418821354
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-1}\).
## [1] -1.7982846779 -3.7564803134 -4.1844989443 -2.4380580230 -2.0573868423
## [6] -0.0910098080 -1.4490056726 -0.6076282073 -1.3496925124 -1.0829677579
## [11] -1.7023873257 -2.5197440501 -2.7197994281 -2.1967896169 -1.7916785770
## [16] -0.4348298378 -1.0473867239 -0.3192377354 -0.3969101190 -0.6227524142
## [21] -0.0924324177 0.9325752668 3.0122584478 2.4454839633 1.4760437919
## [26] -0.0643674725 -0.5461726632 -0.6784655769 0.4243199563 1.9301392423
## [31] 2.1248949046 3.9725615519 4.5550448477 5.3150026476 5.6203880642
## [36] 4.2409849415 4.1294892573 0.9252426580 1.3701387371 -0.5288919768
## [41] -0.5929594395 -0.9757027529 -0.5211792743 -0.5538914045 -0.9307075065
## [46] -0.0187023435 -0.6367046061 -2.2069303322 -2.6390626356 -3.4043427015
## [51] -2.4297860877 0.1390567838 1.6579727568 0.8637556594 2.5428452215
## [56] 0.6022881183 2.1723143763 -0.2438956571 -0.5951906827 -2.9849387954
## [61] -1.7339822802 -4.6654646818 -3.7836495546 -3.9666681459 -4.1325902763
## [66] -3.0262664639 -4.2864109226 -3.7622722444 -4.7151923322 -4.5967456399
## [71] -3.0173218097 -4.2835223541 -3.3062941169 -3.5092919106 -2.6374499698
## [76] -1.5337718084 -1.3312387130 0.7849170050 1.6064013034 3.9292469905
## [81] 1.9596439795 0.9664737813 0.0669866797 -0.1584849418 -0.4141202333
## [86] -0.6312709023 -0.4620127937 0.4949622808 -0.3416039498 -0.9383995764
## [91] -1.4427075147 -2.2610634558 -2.1325223511 -1.1874697279 -1.5071768803
## [96] 0.3218529332 -1.3783213458 0.3085934849 2.4311778976 2.0806680196
## [101] 2.9673723690 1.4530768428 1.9792495800 2.3624139896 1.8422568201
## [106] 1.7614003802 1.9584343073 2.7174696443 2.5529423211 1.4754614153
## [111] 1.3554392078 0.9762882119 2.1519800469 2.0506378942 1.0566362664
## [116] -0.1564765121 -0.6963019933 -2.2441924588 -1.8416334056 -1.8481320229
## [121] -0.9362785485 1.0132172775 -0.1475489089 -1.0352244145 -0.4098905792
## [126] 0.7560315697 3.5576045354 2.1448787201 1.8165941197 -0.5235159857
## [131] -1.1697917262 -1.5151996272 -0.2480426055 0.1076470164 -0.0009816395
## [136] 0.2239496072 -0.3267270455 0.9924955244 -0.8552703690 0.6436579300
## [141] 1.2226656780 3.0756483638 2.7230731085 1.1818983317 -0.2938174133
## [146] 0.2347314183 -1.1966072807 0.3867847283 -1.3827057885 -0.7294145525
## [151] -0.9784002367 -1.2956348622 0.0031996111 -0.3001120910 1.7887425967
## [156] 3.7384623212 2.3458004938 4.0714796600 2.7815006616 3.1478649220
## [161] 1.6664269995 3.2494060618 2.6280455760 5.4679647463 3.5571011493
## [166] 4.5485533388 4.9684033961 5.0280773241 4.6975450886 2.6840203628
## [171] 1.6290009627 0.1999273113 0.5740551176 -0.4256382849 0.5406114019
## [176] 0.7418567933 1.2997851726 1.6025208730 1.6136129632 2.2928430253
## [181] 1.4468904019 1.8612868351 0.8777139291 -0.0610314965 -1.3332268089
## [186] -0.9125263389 0.3581329351 -0.7915369668 0.4955112665 -0.4502244253
## [191] 1.4675554008 3.0521765941 2.0408339742 3.8704090143 3.9605441093
## [196] 3.9360503187 3.5445138311 0.5049588491 0.3915050022 -2.8725640338
## [201] -2.0091667120 -3.0173010664 -1.8381977156 -1.5748868384 -1.2514830640
## [206] -2.5783738178 -3.7484191100 -3.8057155958 -3.0969262848 -1.4586533030
## [211] 0.1531135428 0.8236045641 -0.3790786936 -1.0001442542 -2.2513926710
## [216] -2.5717361239 0.1246023618 -0.5437266688 2.9508487574 1.5184775014
## [221] 2.4469359872 3.3434532407 3.1174139183 2.9978533478 2.4173578196
## [226] 1.5319233546 1.7966371548 1.2212161425 1.9888356266 2.5393974322
## [231] 3.7546275738 1.3043079574 -0.2394983994 -1.7460452432 -0.7708653105
## [236] -0.3849718038 -1.1442085530 -1.5138630876 -2.1968083475 -1.6095317523
## [241] -0.9686538607 -1.8729730474 -2.5265869712 -3.1503634658 -1.6735889734
## [246] -1.2527046715 -0.3854214991 -1.2299702469 -1.3664692859 -1.3620096392
## [251] -0.8698815763 -1.1048884660 -2.4558106193 -3.2188795522 -5.4245906442
## [256] -5.7588922841 -5.9623854154 -5.0454539055 -3.9564322618 -3.7899616180
## [261] -5.3313166036 -4.8885674466 -2.0508739512 -2.0599004072 -1.0821178750
## [266] 0.1666168226 -0.8877977615 0.8502273032 -0.1476428826 -0.4367626603
## [271] -2.1069453839 -4.5465410226 -4.8087456252 -4.2289076874 -1.9178765546
## [276] -2.2804864851 -0.4098672855 -1.6178574433 -0.0137269804 0.0122011943
## [281] 1.4282807605 0.6560697941 0.7567608782 -0.8228605730 -1.1860435575
## [286] -1.8041134735 -0.4466562647 -0.3954967938 2.1487643901 0.9586140341
## [291] 2.2653125680 0.2588626762 1.0329576284 -0.9297517508 -1.2122608974
## [296] -2.3845725749 -5.0393956847
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-2}\).
## [1] -1.3696174977 -1.7982846779 -3.7564803134 -4.1844989443 -2.4380580230
## [6] -2.0573868423 -0.0910098080 -1.4490056726 -0.6076282073 -1.3496925124
## [11] -1.0829677579 -1.7023873257 -2.5197440501 -2.7197994281 -2.1967896169
## [16] -1.7916785770 -0.4348298378 -1.0473867239 -0.3192377354 -0.3969101190
## [21] -0.6227524142 -0.0924324177 0.9325752668 3.0122584478 2.4454839633
## [26] 1.4760437919 -0.0643674725 -0.5461726632 -0.6784655769 0.4243199563
## [31] 1.9301392423 2.1248949046 3.9725615519 4.5550448477 5.3150026476
## [36] 5.6203880642 4.2409849415 4.1294892573 0.9252426580 1.3701387371
## [41] -0.5288919768 -0.5929594395 -0.9757027529 -0.5211792743 -0.5538914045
## [46] -0.9307075065 -0.0187023435 -0.6367046061 -2.2069303322 -2.6390626356
## [51] -3.4043427015 -2.4297860877 0.1390567838 1.6579727568 0.8637556594
## [56] 2.5428452215 0.6022881183 2.1723143763 -0.2438956571 -0.5951906827
## [61] -2.9849387954 -1.7339822802 -4.6654646818 -3.7836495546 -3.9666681459
## [66] -4.1325902763 -3.0262664639 -4.2864109226 -3.7622722444 -4.7151923322
## [71] -4.5967456399 -3.0173218097 -4.2835223541 -3.3062941169 -3.5092919106
## [76] -2.6374499698 -1.5337718084 -1.3312387130 0.7849170050 1.6064013034
## [81] 3.9292469905 1.9596439795 0.9664737813 0.0669866797 -0.1584849418
## [86] -0.4141202333 -0.6312709023 -0.4620127937 0.4949622808 -0.3416039498
## [91] -0.9383995764 -1.4427075147 -2.2610634558 -2.1325223511 -1.1874697279
## [96] -1.5071768803 0.3218529332 -1.3783213458 0.3085934849 2.4311778976
## [101] 2.0806680196 2.9673723690 1.4530768428 1.9792495800 2.3624139896
## [106] 1.8422568201 1.7614003802 1.9584343073 2.7174696443 2.5529423211
## [111] 1.4754614153 1.3554392078 0.9762882119 2.1519800469 2.0506378942
## [116] 1.0566362664 -0.1564765121 -0.6963019933 -2.2441924588 -1.8416334056
## [121] -1.8481320229 -0.9362785485 1.0132172775 -0.1475489089 -1.0352244145
## [126] -0.4098905792 0.7560315697 3.5576045354 2.1448787201 1.8165941197
## [131] -0.5235159857 -1.1697917262 -1.5151996272 -0.2480426055 0.1076470164
## [136] -0.0009816395 0.2239496072 -0.3267270455 0.9924955244 -0.8552703690
## [141] 0.6436579300 1.2226656780 3.0756483638 2.7230731085 1.1818983317
## [146] -0.2938174133 0.2347314183 -1.1966072807 0.3867847283 -1.3827057885
## [151] -0.7294145525 -0.9784002367 -1.2956348622 0.0031996111 -0.3001120910
## [156] 1.7887425967 3.7384623212 2.3458004938 4.0714796600 2.7815006616
## [161] 3.1478649220 1.6664269995 3.2494060618 2.6280455760 5.4679647463
## [166] 3.5571011493 4.5485533388 4.9684033961 5.0280773241 4.6975450886
## [171] 2.6840203628 1.6290009627 0.1999273113 0.5740551176 -0.4256382849
## [176] 0.5406114019 0.7418567933 1.2997851726 1.6025208730 1.6136129632
## [181] 2.2928430253 1.4468904019 1.8612868351 0.8777139291 -0.0610314965
## [186] -1.3332268089 -0.9125263389 0.3581329351 -0.7915369668 0.4955112665
## [191] -0.4502244253 1.4675554008 3.0521765941 2.0408339742 3.8704090143
## [196] 3.9605441093 3.9360503187 3.5445138311 0.5049588491 0.3915050022
## [201] -2.8725640338 -2.0091667120 -3.0173010664 -1.8381977156 -1.5748868384
## [206] -1.2514830640 -2.5783738178 -3.7484191100 -3.8057155958 -3.0969262848
## [211] -1.4586533030 0.1531135428 0.8236045641 -0.3790786936 -1.0001442542
## [216] -2.2513926710 -2.5717361239 0.1246023618 -0.5437266688 2.9508487574
## [221] 1.5184775014 2.4469359872 3.3434532407 3.1174139183 2.9978533478
## [226] 2.4173578196 1.5319233546 1.7966371548 1.2212161425 1.9888356266
## [231] 2.5393974322 3.7546275738 1.3043079574 -0.2394983994 -1.7460452432
## [236] -0.7708653105 -0.3849718038 -1.1442085530 -1.5138630876 -2.1968083475
## [241] -1.6095317523 -0.9686538607 -1.8729730474 -2.5265869712 -3.1503634658
## [246] -1.6735889734 -1.2527046715 -0.3854214991 -1.2299702469 -1.3664692859
## [251] -1.3620096392 -0.8698815763 -1.1048884660 -2.4558106193 -3.2188795522
## [256] -5.4245906442 -5.7588922841 -5.9623854154 -5.0454539055 -3.9564322618
## [261] -3.7899616180 -5.3313166036 -4.8885674466 -2.0508739512 -2.0599004072
## [266] -1.0821178750 0.1666168226 -0.8877977615 0.8502273032 -0.1476428826
## [271] -0.4367626603 -2.1069453839 -4.5465410226 -4.8087456252 -4.2289076874
## [276] -1.9178765546 -2.2804864851 -0.4098672855 -1.6178574433 -0.0137269804
## [281] 0.0122011943 1.4282807605 0.6560697941 0.7567608782 -0.8228605730
## [286] -1.1860435575 -1.8041134735 -0.4466562647 -0.3954967938 2.1487643901
## [291] 0.9586140341 2.2653125680 0.2588626762 1.0329576284 -0.9297517508
## [296] -1.2122608974 -2.3845725749
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-3}\).
## [1] 0.8628549
Nilai korelasi antara \(Y_t\) dengan \(Y_{t-1}\) dari hasil simulasi cukup mendekati dengan teori, yaitu:
\[ \rho_1 = \frac{(1 - \phi_2)\theta_1 + (\phi_1 - \phi_2)\theta_2 + \phi_1(1 + \theta_1^2 + \theta_2^2)} {(1 - \phi_2)(1 + \theta_1^2 + \theta_2^2) + 2(\phi_1 - \phi_2)\theta_1\theta_2} = \frac{(1 - 0.2)(0.4) + (0.5 - 0.2)(0.6) + 0.5(1 + 0.4^2 + 0.6^2)} {(1 - 0.2)(1 + 0.4^2 + 0.6^2) + 2(0.5 - 0.2)(0.4)(0.6)} = \frac{1.26}{1.36} \approx 0.926 \]
## [1] 0.7616306
Nilai korelasi antara \(Y_t\) dengan \(Y_{t-2}\) dari hasil simulasi cukup mendekati dengan teori, yaitu:
\[ \rho_2 = \frac{(\phi_1 + \theta_1\theta_2)(1 - \phi_2) + (\phi_1^2 - \phi_2)(1 + \theta_1^2 + \theta_2^2) + \theta_2(\theta_1 + \phi_1\theta_2)} {(1 - \phi_2)(1 + \theta_1^2 + \theta_2^2) + 2(\phi_1 - \phi_2)\theta_1\theta_2} = \frac{(0.5 + 0.4 \times 0.6)(0.8) + (0.5^2 - 0.2)(1 + 0.4^2 + 0.6^2) + 0.6(0.4 + 0.5 \times 0.6)} {(1-0.2)(1 + 0.4^2 + 0.6^2) + 2(0.5 - 0.2)(0.4)(0.6)} = \frac{1.088}{1.36} = 0.8 \]
## [1] 0.5721462
Nilai korelasi antara \(Y_t\) dengan \(Y_{t-3}\) dari hasil simulasi cukup mendekati teori, yaitu:
\[ \rho_3 = \phi_1 \rho_2 + \phi_2 \rho_1 = (0.5)(0.8) + (0.2)(0.926) = 0.585 \]
Kesimpulan: Meskipun EACF mengindikasikan ARMA(1,2), pola ACF dan PACF yang tails off serta nilai autokorelasi empiris dengan nilai teoritis yang tidak berbeda jauh menunjukkan bahwa model ARMA(2,2) merupakan representasi yang konsisten untuk proses bangkitan. Perbedaan kecil pada hasil identifikasi tampaknya muncul akibat variabilitas sampel dan sensitivitas EACF, bukan karena inkonsistensi struktural model.